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Full text of "An elementary treatise on quaternions"

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Cornell University Library 
QA 196.T13 1873 

An elementary treatise on quaternions 

3 1924 001 570 971 





SonDon : Cambridge warehouse, 


©amfiriDgc: deighton, bell, and co. 





P. G. TAIT, M.A. 



nayav aevaov <f)V(rfas pif^iijiar txovaav. 



[All Rights reserved,} 

The original of tiiis book is in 
tine Cornell University Library. 

There are no known copyright restrictions in 
the United States on the use of the text. 



To THE first edition of this work, published in 1867, the following 
was prefixed : — 

' The present work was commenced in 1859, while I was a Pro- 
fessor of Mathematics, and far more ready at Quaternion analysis 
than I can now pretend to be. Had it been then completed I 
should have had means of testing its teaching capabilities, and of 
improving it, before publication, where found deficient in that 

' The duties of another Chair, and Sir W. Hamilton's wish that 
my volume should not appear till after the publication of his JSle- 
ments, interrupted my already extensive preparations. I had worked 
out nearly all the examples of Analytical Geometry in Todhunter's 
Collection, and I had made various physical applications of the 
Calculus, especially to Crystallography, to Geometrical Optics, and 
to the Induction of Currents, in addition to those on Kinematics, 
Electrodynamics, Fresnel's Wave Surface, &c., which are reprinted 
in the present work from the Quarterly Mathematical Journal and 
the Proceedings of the Royal Society of Edinburgh. 

' Sir W. Hamilton, when I saw him but a few days before his 
death, urged me to prepare my work as soon as possible, his being 
almost ready for publication. He then expressed, more strongly 
perhaps than he had ever done before, his profound conviction of 
the importance of Quaternions to the progress of physical science ; 
and his desire that a really elementary treatise on the subject should 
soon be published. 


' I regret that I have so imperfectly fulfilled this last request of 
my revered friend. When it was made I was already engaged, 
along with Sir W. Thomson, in the laborious work of preparing 
a large Treatise on Natural Philosophy. The present volume has 
thus been written under very disadvantageous circumstances, espe- 
cially as I have not found time to work up the mass of materials 
which I had originally collected for it, but which I had not put 
into a fit state for publication. I hope, however, that I have to 
some extent succeeded in producing a thoroughly elementary work, 
intelligible to any ordinary student; and that the numerous ex- 
amples I have given, though not specially chosen so as to display 
the full merits of Quaternions, will yet sufficiently shew their admir- 
able simplicity and naturalness to induce the reader to attack the 
Lectures and the Elements ; where he will find, in profusion, stores 
of valuable results, and of elegant yet powerful analytical investiga- 
tions, such as are contained in the writings of but a very few of the 
greatest mathematicians. For a succinct account of the steps by 
which Hamilton was led to the invention of Quaternions, and for 
other interesting information regarding that remarkable genius, I 
may refer to a slight sketch of his life and works in the North 
British Review for September 1866. 

' It will be found that I have not servilely followed even so great 
a master, although dealing with a subject which is entirely his 
own. I cannot, of course, tell in every case what I have gathered 
from his published papers, or from his voluminous correspondence, 
and what I may have made out for myself. Some theorems and 
processes which I have given, though wholly my own, in the sense 
of having been made out for myself before the publication of the 
Elements, I have since found there. Others also may be, for 1 have 
not yet read that tremendous volume completely, since much of it 
bears on developments unconnected with Physics. But I have 
endeavoured throughout to point out to the reader all the more 
important parts of the work which I know to be wholly due to 
Hamilton. A great part, indeed, may be said to be obvious to any 
one who has mastered the preliminaries ; still I think that, in the 


two last Chapters especially, a good deal of original matter will be 

' The volume is essentially a working one, and, particularly in the 
later Chapters, is rather a collection of examples than a detailed 
treatise on a mathematical method. I have constantly aimed at 
avoiding too great extension ; and in pursuance of this object have 
omitted many valuable elementary portions of the subject. One of 
these, the treatment of Quaternion logarithms and exponentials, I 
greatly regret not having given. But if I had printed all that 
seemed to me of use or interest to the student, I might easily have 
rivalled the bulk of one of Hamilton's volumes. The beginner is 
recommended merely to read the first five Chapters, then to work 
at Chapters VI, VII, VIII (to which numerous easy Examples are 
appended). After this he may work at the first five, with their 
(more difficult) Examples ; and the remainder of the book should 
then present no difficulty. 

' Keeping always in view, as the great end of every mathematical 
method, the physical applications, I have endeavoured to treat the 
subject as much as possible from a geometrical instead of an analy- 
tical point of view. Of course, if we premise the properties of i,j, k 
merely, it is possible to construct from them the whole system* ; 
just as we deal with the imaginary of Algebra, or, to take a closer 
analogy, just as Hamilton himself dealt with Couples, Triads, and 
Sets. This may be interesting to the pure analyst, but it is repulsive 
to the physical student, who should be led to look upon i, _/, k from 
the very first as geometric realities, not as algebraic imaginaries. 

' The most striking peculiarity of the Calculus is that muUipli- 
cation is not generally commutative, i.e. that qr is in general different 
from rq, r and q being quaternions. Still it is to be remarked that 
something similar is true, in the ordinary coordinate methods, of 
operators and functions : and therefore the student is not wholly 
unprepared to meet it. No one is puzzled by the fact that log.cos.a; 

* This has been done by Hamilton himself, as one among many methods he has 
employed ; and it is also the foundation of a memoir by M. AU^gret, entitled Esmi 
sv/r le Calcul des Quaternions (Paris, 1862). 

viii PREFACE. 

is not equal to cos.log.a?, or that v/ j- is not equal to -^s/V' 

Sometimes, indeed, this rule is most absurdly violated, for it is 
usual to taJie cos^a; as equal to (cos xY, while cos-^a; is not equal to 
(cos «)"■'. No such incongruities appear in Quaternions j but what 
is true of operators and functions in other methods, that they are 
not generally commutative, is in Quaternions true in the multipli- 
cation of (vector) coordinates. 

' It will be observed by those who are acquainted with the Cal- 
culus that I have, in many cases, not given the shortest or simplest 
proof of an important proposition. This has been done with the 
view of including, in moderate compass, as great a variety of 
methods as possible. With the same object I have endeavoured to 
supply, by means of the Examples appended to each Chapter, hints 
(which will not be lost to the intelligent student) of farther develop- 
ments of the Calculus. Many of these are due to Hamilton, who, 
in spite of his great originality, was one of the most excellent 
examiners any University can boast of. 

' It must always be remembered that Cartesian methods are mere 
particular cases of Quaternions, where most of the distinctive fea- 
tures have disappeared; and that when, in the treatment of any 
particular question, scalars have to be adopted, the Quaternion 
solution becomes identical with the Cartesian one.. Nothing there- 
fore is ever lost, though much is generally gained, by employing 
Quaternions in preference to ordinary methods. In fact, even when 
Quaternions degrade to scalars, they give the solution of the most 
general statement of the problem they are applied to, quite inde- 
pendent of any limitations as to choice of particular coordinate 

'There is one very desirable object which such a work as this 
may possibly fulfil. The University of Cambridge, while seeking 
to supply a real want (the deficiency of subjects of examination for 
mathematical honours, and the consequent frequent introduction of 
the wildest extravagance in the shape of data for " Problems "), is 
in danger of making too much of such elegant trifles as Trilinear 


Coordinates, while gigantic systems like Invariants (which, by the 
way, are as easily introduced into Quaternions as into Cartesian 
methods) are quite beyond the amount of mathematics which even 
the best students can master in three years' reading. One grand 
step to the supply of this want is, of course, the introduction into 
the scheme of examination of such branches of mathematical physics 
as the Theories of Heat and Electricity. But it appears to me that 
the study of a mathematical method like Quaternions, which, while 
of immense power and comprehensiveness, is of extraordinary sim- 
plicity, and yet requires constant thought in its applications, would 
also be of great benefit. With it there can be no " shut your eyes, 
and write down your equations," for mere mechanical dexterity of 
analysis is certain to lead at once to error on account of the novelty 
of the processes employed. 

'The Table of Contents has been drawn up so as to give the 
student a short and simple summary of the chief fundamental for- 
mulae of the Calculus itself, and is therefore confined to an analysis 
of the first five [and the two last] chapters. 

' In conclusion, I have only to say that I shall be much obliged 
to any one, student or teacher, who will point out portions of the 
work where a difficulty has been found ; along with any inaccuracies 
which may be detected.. As I have had no assistance in the revision 
of the proof-sheets, and have composed the work at irregular in- 
tervals, and while otherwise laboriously occupied, I fear it may 
contain many slips and even errors. Should it reach another edition 
there is no doubt that it will be improved in many important par- 

To this I have now to add that I have been equally surprised 
and delighted by so speedy a demand for a second edition — and the 
more especially as I have had many pleasing proofs that the 
work has had considerable circulation in America. There seems 
now at last to be a reasonable hope that Hamilton's grand in- 
vention will soon find its way into the working world of science, 
to which it is certain to render enormous services, and not be laid 


aside to be unearthed some centuries hence by some grubbing 

It can hardly be expected that one whose time is mainly en- 
grossed by physical science, should devote much attention to the 
purely analytical and geometrical applications of a subject like this ; 
and I am conscious that in many parts of the earlier chapters I 
have not fully exhibited the simplicity of Quaternions. I hope, 
however, that the corrections and extensions now made, especially 
in the later chapters, will render the work more useful for my chief 
object, the Physical Applications of Quaternions, than it could have 
been in its first crude form. 

I have to thank various correspondents, some anonymous, for 
suggestions as well as for the detection of misprints and slips of 
the pen. The only absolute error which has been pointed out to 
me is a comparatively slight one which had escaped my own notice : 
a very grave blunder, which I have now corrected, seems not to 
have been detected by any of my correspondents, so that I cannot 
be quite confident that others may not exist. 

I regret that I have not been able to spare time enough to re- 
write the work ; and that, in consequence of this, and of the large 
additions which have been made (especially to the later chapters), 
the whole will now present even a more miscellaneously jumbled 
appearance than at first. 

It is well to remember, however, that it is quite possible to 
make a book too easy reading, in the sense that the student may 
read it through several times without feeling those difficulties 
which (except perhaps in the case of some rare genius) must 
attend the acquisition of really useful knowledge. It is better to 
have a rough climb (even cutting one's own steps here and there) 
than to ascend the dreary monotony of a marble staircase or a 
well-made ladder. Royal roads to knowledge reach only the par- 
ticular locality aimed at — and there are no views by the way. 
It is not on them that pioneers are trained for the exploration of 
unknown regions. 

But I am happy to say that the "possible repulsiveness of my 


early chapters cannot long- be advanced as a reason for not at- 
tacking this fascinating subject. A still more elementary work 
than the present will soon appear, mainly from the pen of my 
colleague Professor Kelland. In it I give an investigation of 
the properties of the linear and vector function, based directly 
upon the Kineinatics of Homogeneous Strain, and therefore so 
different in method from that employed in this work that it may 
prove of interest to even the advanced student. 

Since the appearance of the first edition I have managed (at least 
partially) to effect the application of Quaternions to line, surface, 
and volume integrals, such as occur in Hydrokinetics, Electricity, 
and Potentials generally. I was first attracted to the study of 
Quaternions by their promise of usefulness in such applications, 
and, though I have not yet advanced far in this new track, I have 
got far enough to see that it is certain in time to be of incalculable 
value to physical science. I have given towards the end of the 
work all that is necessary to put the student on this track, which 
will, I hope, soon be followed to some purpose. 

One remark more is necessary. I have employed, as the positive 
direction of rotation, that of the earth about its axis, or about the 
sun, as seen in our northern latitudes, i.e. that opposite to the direc- 
tion of motion of the hands of a watch. In Sir W. Hamilton's 
great works the opposite is employed. The student will find no 
difficulty in passing from the one to the other ; but, without pre- 
vious warning, he is liable to be much perplexed. 

With regard to notation, I have retained as nearly as possible 
that of Hamilton, and where new notation was necessary I have 
tried to make it as simple and as little incongruous with Hamil- 
ton's as possible. This is a part of the work in which great care 
is absolutely necessary; for, as the subject gains development, 
fresh notation is inevitably required ; and our object must be to 
make each step such as to defer as long as possible the revolution 
which must ultimately come. 

Many abbreviations are possible, and sometimes very useful in 
private work ; but, as a rule, they are un suited for print. Every 


analyst, like every short-hand writer, has his own special con- 
tractions ; but, when he comes to publish his results, he ought 
invariably to put such devices aside. If all did not use a com- 
mon mode of public expression, but each were to print as he is 
in the habit of writing for his own use, the confusion would be 
utterly intolerable. 

Finally, I must express my great obligations to my friend 
M. M. U. Wilkinson of Trinity College, Cambridge, for the care 
with which he has read my proofs, and for many valuable sug- 

P. G. TAIT. 

College, EDnfBUROH, 
Octoher 1873. 


Chapter I. — Vectoks and their Composition 1-22 

Sketch of the attempts made to represent geometrically the unaginary of 

algebra. §§ 1-13. 
De Moivre's Theorem interpreted in plane rotation. § 8. 
Curious speculation of Servois. §11. 

Elementary geometrical ideas connected 'with relative position. § 15. 
Definition of a Vbctoe. It may be employed to denote translation. § 16. 
Expression of a vector by one symbol, containing implicitly three distinct 

numbers. Extension of the signification of the symbol = . § IS. 
The sign + defined in accordance with the interpretation of a rector as 

representing translation. § 19. 
Definition of - . It simply reverses a vector. § 20. 

Triangles and polygons of vectors, analogous to those of forces and of simul- 
taneous velocities. § 31. 
When two vectors are paralkl we have 

a = xp. § 22. 
Any vector whatever may be expressed in terms of three distinct vectors, 

which are not coplana, by the formula 
p = xa+yP+zy, 

which exhibits the three numbers on which the vector depends. § 23. 
Any vector in the same plane with a and /S may be written 

p = xa+yp. §24. 
The equation 'sr = p, 

between two vectors, is equivalent to three distinct equations among 

numbers. § 25. 
The Oonmmtative and Associative Laws hold in the combination of vectors by 

the signs + and - . § 27. 
The equation p = »/S, 

where ^ is a variable, and p a fixed, vector, represents a line drawn 

through the origin parallel to j3. 

p = O + K/S 

is the equation of a line drawn through the extremity of a and parallel 
to jS, § 28. 

p = ya+x§ 
represents the plane through the origin parallel to a and p. § 29. 


The condition that p, a, /3 may terminate in the same line is 
p/) + jo + r/S = 0, 
subject to the identical relation 

Similarly pp + qa + r§ + ay = 0, 

with p + q^ + r-\rs = 0, 

is the condition that the extremities of four vectors lie in one plane. § 30. 
Examples with solutions. § 81. 

Differentiation of a vector, when given as a function of one number. §§ 32-38. 
If the equation of a curve be 

p = 4>{s) 

where s is the length of the arc, dp is a vector tangent to the curve, and 

its length is ds. §§ 38, 39. 
Examples with solutions. §§ 40-44. 

Examples to Chapter 1 22-24 

Chapter II. — Products and Quotients of Vectors . . . 25-46 

Here we begin to see what a quaternion is. When two vectors are parallel 
their quotient is a number. §§ 45, 46. 

When they are perpendicular to one another, their quotient is a vector per- 
pendicular to their plane. § 47, 72. 

When they are neither parallel nor perpendicular the quotient in general 
involves fovtr distinct numbers — and is thus a Quatbbnion. § 47. 

A quaternion regarded as the operator which turns one vector into another. 
It is thus decomposable into two factors, whose order is indifferent, the 
stretching factor or Tenbob, and the turning factor or Versob. These 
are denoted by Tq, and Uq. § 48. 

The equation /3 = j a 


gives = It or /3a~' = g, hiit not in general 

nr^^ = q. §49. 
q or j3a~' depends only on the relative lengths, and directions, of jS and a. 

Reci^ocal of a quaternion defined, 

2=-g,ves-orgi = -, 

y-2-' = -^. U.q-^ = {Uq)-\ §51. 

Definition of the Conjugate of a quaternion, 
and qKq = Kq.q = [Tqf. § 52. 
Eepresentation of versors by arcs on the unit-sphere. § 53. 
Versor multiplication illustrated by the composition of arcs. § 54. 
Proof that K{qr) = Kr . Kq. § 55. 

Proof of the Associative Law of Multiplication 

p.qr=^pq.r. §§57-60. 
[Digression on Spherical Conies. § 59'.] 


Quaternion addition and subtraction are commutative. § 61. 
Quaternion multiplication and division are disti-^uiive. § 62. 
CompoEdtion of quadrantal veraors in planes at right angles to each other. 
Calling them i, j, k, we have 
i'=f=k^= - 1, »)■= -ji = h, jh= -hj=i, K= -ilc=j, 
yh=-l. §§64-71. 
A unit-vector, when employed as a factor, may be considered as a quadrautal 
versor whose plane is perpendicular to the vector, Hence the equations 
just written are true of any set of rectangular unit-vectors i, j, Js. § 72. 
The product, and the quotient, of two vectors at right angles to each other is 
a third perpendicular to both. Hence 
Ka = -a, 
and {Ta)^ = aKa=-a'. §73. 
Every versor may be expressed as a power of some unit-vector. § 74. 
Every quaternion may be expressed as a power of a vector. § 73. 
The Index Law is true of quaternion multiplication and division. § 76. 
Quaternion considered as the sum of a SOALAB and Yeoiob. 

q = ^ = x+y = Si + Vi. §77., 

Proof that SKq = Sq, YKq = -7q, §79. 

Quadrinomial expression for a quaternion 

q = w+ix+jy + Jcz. 
An equation between quaternions is equivalent to four equations between 

numbers (or scalars). § 80. 
Second proof of the distributive law of multiplication. § 81. 
Algebraic determination of the constituents of the product and quotient of two 

vectors. §§ 82-84. 
Second proof of the associative law of multiplication. § 85. 
Proof of the formulae SajS = S^a, 
FajS = - rpa, 
o/S = K^a, 
S.qrs = S.rsq = S.sqr, 

S. a/Sv = S.pya = iS'.70jS = - S^ ayp = &c. §§ 86-89. 
Proof of the formulae 

V.aVpy = ySaP-pSya, 
V. 0JS7 = aSpy - pSya + 7/S0/S; 
7.0/87= ^-Y/So, 
V. FajS Vyd = o<S'.;875 - 185.07S, 
= SS.aPy-yS.a^S, 
SS.aPy = aS.pyS + pS.yaS + yS.apS, 

= VapSyS+ r§ySad+ VyaSpS. §§ 90-92. 
Hamilton's proof that the product of two parallel vectors must be a scalar, and 
that of perpendicular vectors, a vector; if quaternions are to deal with 
space indifferently in all directions. § 93. 

Examples to Chaptek II 46, 47 


Chapter III. — Interpretations and Transformations of 

Quaternion Expressions 48-67 

If 6 be the angle between two vectors, a and j9, we have 

S^ = ^cos e, SaB = - TaT^ cosff, 

o xo 

a Ta 
Applications to plane trigonometry. §§ 94-97. 

shews that o is perpendicular to jS, while 
Fo/3 = 0, 

shews that a and p are parallel. 


is the volume of the parallelepiped three of whose conterminous edges are 

a, jS, 7. Hence S.aPy = 

shews that a, j3, 7 are coplanar. 

Expression of S. apy as a determinant. §§ 98-102. 
Proof that {Tg)' = {Si)'+ {TVq)", 

and T{qr) = Tq, Tr. % 103, 

Simple propositions in plane trigonometry. § 104. 
Proof that - apa~^ ia the vector reflected ray, when j3 is the incident ray and o 

normal to the reflecting surface. § 105. 
Interpretation of 0/87 when it is a vector. § 106. 
Examples of variety in simple transformations. § 107. 
Introduction to spherical trigonometry. §§ 108-113. 

Bepreaentation, graphic, and by quaternions, of the spherical excess. §g 114, IIS. 
Loci represented by different equations — points, lines, surfaces, and solids. 

§§ 116-119. 
Proof that r-i (rV)* g-i = U(rq + KrKq). § 120. 

Proof of the transformation 

(Sv^pf + (S^pf + (Sypf = i^^^y, 

-''^ M«-}=^"(v55l>«-v^>7). ^121.122. 

BlQUATEENlONS. §§ 123-125. 

Convenient abbreviations of notation. §§ 126, 127. 

Examples to Chaptbe III 68-70 

Chapter IV. — Differentiation op Quaternions .... 71-76 
Definition of a dififerential, 

where dq is any quaternion whatever. 
We may write dFq =f{q, dq), 

where / is linear and homogeneous in dq; but we cannot generally write 
dFq = f{q)dq. §§128-131. 


Definition of the differential of aTTunction of more qnatemions ttan one. 
d(qr) = qdr + dq.r, but not generally d($r) = qdr + rdq. § 132. 

Proof that ^ = S^, 

Tp p 

^=F^,&c. §133. 
Up p 

Successive differentiation; Taylor's theorem. §§ 134, 135. 

If the equation of a surface be 

-P0>) = c, 

the differential may be written 

Svdp = 0, 
where >< is a vector normal to the surface. § 137. 
Examples to Chaptbe IV 76 

Chaptee V. — The Solution op Equations op the First Degree. 


The most general equation of the first degree in an unknown quaternion q, 

may be written 2 V. aqb + S .cq = d, 

where a, h, c, d are given quaternions. Elimination of ;S'}, and reduction 

to the vector equation 

<1>P = S. aSPp = y. |§ 138, 139.' : -^ 
General proof that ^'p is expressible as a linear function of p, <pp, and <p'p. 

Value of <l> for an ellipsoid, employed to illustrate the general theory. 

§§ 141-143. 
Hamilton's solution of (pp = y. 

If we write Sa<j>p = Sp(p'a, 

the functions <p and ^' are said to be conjugate, and 
m^-^V\ii = T<t>'\<t>'p.. 

Proof that m, whose value may be written as 
S .<p'K<p! fup'v 

is the same for all values of \, n,v. §§ 144-146. 
Proof that if »n^ = m + jHi jr + 7»j jr» + j', 

S (\<p'iup'v + f'Xfup'v + <p'\<p'ia/) 

where m^ = 


8 (X^0V + <l>'\iiv + K(p'nv) 

' S.\iiv 

(which, like m, are Invarianti,) 
then mg (<p + g)~^ VXn = (m^-' -k-gx + f) ^^f- 

Also that X = '»a— ■/>> 

whence the final form of solution 

m<p-^ = m.i-mi<p + <p''. §§147,148. 
Examples. § 149-161. 


xviii CONTENTS. 

The fundamental cubic 

(/I'-mjif' + m.^-m = (<f-£rj) (^-ffi)("?'-ff.) = 0. 
When is its own conjugate, the roots of the cubic are real ; and the 
equation ^Plip. = 0> 

or (.<p-g) P = 0, 
is satisfied by a set of three real and mutually perpendicular vectors. 
Geometrical interpretation of these results. §§ 162-166. 
Proof of the transformation 

i>p =fp + hV. (i + ek)'f (i—ek) 
where (<^— ffi)* = 0, 

C — ) 

Another transformation is 

(pp = aaVap + bPSPp. §§167-169. 
Other properties of i^. Proof that 

Sp(<t> + g)~*P = 0) and Sp (<p + h)~^p = 
represent the same surface if 

mSp(p~^p = ghp^. 
Proof that when ip is not self-conjugate 

ipp = (p'p + Vep. 
Proof that, if q = a(pa + 0(p0 + 71^7, 

where a, P, 7 are any rectangular unit-vectors whatever, we have 

Sq— — TOj, Vq = f. 

This quaternion can be expressed in the important form 
2 = v#. §§ 170-174. 
Degrees of indeterminateness of the solution of a quaternion equation — 

Examples. §§ 176-179. 
The linear function of a quaternion is given by a symbolical biquadratic. 

Particular forms of linear equations. §§ 181-183. 
A quaternion equation of the mth degree in general involves a scalar equation 

of degree m*. § 184. 
Solution of the equation ^ = qa + T>. §185. 

Examples to ChaptebV 101-103 

Chapter VI. — Gbometei of the Straight Line and Plane , 

Examples to Chafteb VI 117-119 

Chapter VII. — ^The Sphere and Cyclic Cone . . . 120-132 
EXAMPLES TO Chapteb VII 132-134 

Chapter VIII. — Surfaces of the Second Order . . 135-151 
Examples xo Chapter VIII 151-154 


Chapter IX. — Geometry op (?urves and Surfaces . 155-186 
Examples to Chapter IX 187-194 

Chapter X. — Kinematics 195-218 

If p = 0< be the vector of a moving point in terms of the time, p is the 
vector velocity, and p the vector acceleration. 

(T = p = ((>'(<) is the equation of the Hodograph. 
p = vp' + v'p" gives the normal and tangential accelerations. 
Vpp = if acceleration directed to a point, whence Tpp = y. 
Examples. — Planetary acceleration. Here the equation is 

given Vp^ — y ; whence the hodograph is 
p = ty~'^—iiUp.y~\ 

and the orbit is the section of 

j«r/. = Sf(7='£->-p) 

by the plane Syp = 0. 

Epitrochoids, &c. §| 336-348. 
Rotation of a rigid system. Composition of rotations. The operator 5s( )q—^ 

turns the system it is applied to through 2 b times the angle of g, about 

the axis of q. If the position of a system at time t is derived from the 

initial position by j ( ) 2~', the instantaneous axis is 
€ = 2Vqg-^. §§ 349-359. 
Homogeneous strain. Criterion of pure strain. Separation of the rotational 

jrom the pure part. Extraction of the square root of a strain. A strain 

^ is equivalent to a, pure strain V*^'^ followed by the rotation — - — . 

Simple Shear. §§ 360-367. '^'P''!' 

Displacements of systems of points. Consequent condensation and rotation. 

Preliminary about the use of V. §§ 368-371. 
Moment of inertia. § 372. 

Examples to Chapter X 218-221 

Chapter XI. — Physical Applications 222-288 

Condition of equilibrium of -a rigid system is 'SS.PSa = 0, where j8 is a vector 
force, a its point of application. Hence the usual six equations in the 
form 2j8 = 0, SVa0 = 0. Central axis, &c. §§ 373-378. 
For the motion of a rigid system 

SS(md-P)Sa = 0, 
whence the usual forms. The equation 

2j= q<p-^{q-^yq), 
where y is given in terms of t and q if forces act, but is otherwise constant, 
contains the whole theory of the motion of a rigid body with one point 
fixed. Reduction to the ordinary form 

dt dm dx _dy _ dz 
"2 W~X~T~ ~Z' 
Here, if no forces act, W, X, T, Z are homogeneous functions of the third 
degree in w, x, y, z. Equation for precession. §§ 379-401. 
General equation of motion of simple pendulum. Foucault's pendulum. 
§§ 402-406. 



Problem on reflecting surfaces. § 406. 

Freanel's Theory of Double Eefraction. Various fonns of the equation of 

Fresnel's Wave-surface ; 
S.p(.f-p»)-V = -l. T(p-'-<l>-')-ip = 0, l:=-pp' + (,T±S)VKpViip, 

The conical cusps and circles of contact. Lines of vibration, &c. §§ 

Electrodynamics. The vector action of a closed circuit on an element of 

current o, is proportional to Vai0 where 

^ rVada CdUa 

the integration extending round the circuit. Mutual action of two closed 
circuits, and of solenoids. Mutual action of magnets. Potential of a 
closed circuit. Magnetic curves. §§ 428-448. 

Physical applications of 

, d . d , i 
dx dy dz 
Effect of V on various functions of p. = kc +jy + kz. 

V/)=-3, VTp=V'p, vTJp = —~. V8ap=-a, v7ap=2a. 

Applications of the theorem 

S.SpV^ = SS.oV^ . §§ 449-457. 
Jp» Ip 

Farther examination of the use of V as applied to displacements of groups of 

points. Proof of the fundamental theorem for comparing an integral 

over a closed surface with one through its content 


Hence Green's Theorem. Limitations and ambiguities. §§ 458-476. 

Similar theorem for double and single integrals 

fS.adp =//S.UV7ads. 

Applications of these to distributions of magnetism, and to Ampere's 

Directrice. Also to the Stress-function. §5 477-491. 

e-S<rV/(p)= f(p + <r). 

Applications and consequences. Separation of symbols of operation, and 

their treatment as quantities. §§ 492-495. 

Applications of V in connection with the Calculus of Variations. If 

A =/QTdp, SA=0 gives ^(,Qp')-vQ = 0. 


Applications to Varying Action, Brachistochrones, Catenaries. §§ 496-504. 
Thomson's Theorem that there is one and but one solution of 
S.VCe'Vit) = 4irr. §505. 


Page 102, line 20, for ^p—tpipp read <j>if/p—</«l>'p. 




1,] For more than a century and a half the geometrical re- 
presentation of the negative and imaginary algebraic quantities, 
— 1 and a/— 1, or, as some prefer to write them, — and — *, has 
been a favourite subject of speculation with mathematicians. The 
essence of almost all of the proposed processes consists in em- 
ploying such expressions to indicate the direction, not the length, 
of lines. 

2.] Thus it was long ago seen that if positive quantities were 
measured o£F in one direction along a fixed line, a useful and lawful 
convention enabled us to express negative quantities of the same 
kind by simply laying them off on the same line in the opposite 
direction. This convention is an essential part of the Cartesian 
method, and is constantly employed in Analytical Geometry and 
Applied Mathematics. 

3.] WaUis, towards the end of the seventeenth century^ proposed 
to represent the impossible roots of a quadratic equation by going 
out of the line on which, if real, they would have been laid off. 
His construction is equivalent to the consideration of v — 1 as a 
directed unit-line perpendicular to that on which real quantities 
are measured. 

4. J In the usual notation of Analytical ^Geometry of two 
dimensions, when rectangular axes are employed, this amounts 
to reckoning each unit of length along Oy as +v— 1, and on 
Oy' as — V — 1 ; whUe on Ox each unit is +1, and on Oaf it is 



— 1 . If we look at these four lines in circular order, i. e. in the 
order of positive rotation (opposite to that of the hands of a watch), 
they give r—r _ _ y3"i 

In this series each expression is derived from that which precedes 
it by multiplication by the factor v— 1. Hence we may consider 
-v/— 1 as an operator, analogous to a handle perpendicular to the 
plane of ay, whose effect on any line is to make it rotate (positively) 
about the origin through an angle of 90°. 

5.] In such a system^ a point is defined by a single imaginary 
expression. Thus a + b v — 1 may be considered as a single quan- 
tity, denoting the point whose coordinates are a and b. Or, it may 
be used as an expression for the line joining that point with the 
origin. In the latter sense, the expression a + b \/—l implicitly 
contains the direction, as well as the length, of this line ; since, as 

we see at once, the direction is inclined at- an angle tan^^- to the 
axis oi X, and the length is \/a^ + J^. 

6.] Operating on this symbol by the factor V— 1, it becomes 

— 3-|-a\/— 1 ; and now, of course, denotes the point whose x avAy 
coordinates are —b and a ; or the line joining this point with the 
origin. The length is still Va^ + b"^, but the angle the line makes 

with the axis of a; is tan~^ (~ 7") ' 'w^^'ich is evidently 90° greater 

than before the operation. 

7.3 De Moivre's Theorem tends to lead us still farther in the 
same direction. In fact, it is easy to see that if we use, instead 
of >/— 1, the more general factor cosa+ ^/— 1 sin a, its effect on 
any line is to turn it through the (positive) angle a in the plane 
oix,y. [Of course the former factor, 'J —I, is merely the par- 
ticular case of this, when a = - •! 

2 -■ 

Thus (cos a -I- \/ — 1 sina) (a + ^-s/— 1) 

= a cos o — 5 sina-l- \/— 1 (asino + J cos a), 

by direct multiplication. The reader will at once see that the new 
form indicates that a rotation through an angle a has taken place, 
if he compares it with the common formulae for turning the co- 
ordinate axes through a given angle. Or, in a less simple manner, 
thus — 

Length =\/(a coso— 6sina)^ + (asina + 5cosa)^ 
= \/a'^ -I- b'^ as before. 


Inclination to axis of a; j 

, tan a-\ — 

, _, « sin a + cos a , , a 

= tan '■ j-^ — = tan-i = 

a cos a— sin a o 
I 1 tana 

= a + tan~i - • 

8.] We see now, as it were, wA^ it happens that 

(cos a 4- V — 1 sin a)™ = cos »ia + /^/^^^ sin ma. 

In fact, the first operator produces m successive rotations in the 

same direction, each through the angle a ; the second, a single 

rotation through the angle ma. 

9.] It may be interesting, at this stage, to anticipate so far as to 

state that a Quaternion can, in general, he put under the form 

N {cos d + -ay sin 6), 

where iV" is a numerical quantity, 8 a real angle, and 

This expression for a quaternion bears a very close analogy to the 
forms employed in De Moivre's Theoreili ; but there is the essential 
difference (to which Hamilton's chief invention referred) that -sr, 
is not the algebraic v — 1, but may be an^ directed unit-line what- 
ever in space. 

10.] In the present century Argand, Warren, and others, extended 
the results of WalHs and De Moivre. They attempted to express 
as a line the product of two lines each represented by a symbol 
such as a + J v^— 1. To a certain extent they succeeded, but sim- 
plicity was not gained by their methods, as the terrible array of 
radicals in Warren's Treatise suflBciently proves. 

11.] A very curious speculation, due to Servois and published 
in 1813 in Gergonne's Annates, is the only one, so far as has 
been discovered, in which the slightest trace of an anticipation of 
Quaternions is contained. Endeavouring to extend to space the 
form a + J\/— 1 for the plane, he is guided by analogy to write for 
a directed unit-line in space the form 

p cos a + §■ cos /3 + r cos y, 
where a, p, y are its inclinations to the three axes. He perceives 
easily that j9, q, r must be non-reals : but, he asks, " seraient-elles 
imaginaires reductibles a la forme generale A-\-B '^ — I ?" This 
he could not answer. In fact they are the i, j, k of the Quaternion 
Calculus. (See Chap. II.) 

12.] Beyond this, few attempts were made, or at least recorded, in 
earlier times, to extend the principle to space of three dimensions ; 

B a 

4 QUATERNIONS. [ 1 3- 

and, though many such have been made within the last forty 
years, none, with the single exception of Hamilton's, have 
resulted in simple, practical methods; all, however ingenious, 
seeming to lead at once to processes and results of fearful com- 

For a lucid, complete, and most impartial statement of the 
claims of his predecessors in this field we refer to the Preface to 
Hamilton's Lectures on Quaternions. 

13.] It was reserved for Hamilton to discover the use of -n/ — 1 
as a geometric realitij, tied down to no particular direction in space, 
and this use was the foundation of the singularly elegant, yet 
enormously powerful, Calculus of Quaternions. 

While all other schemes for using -s/^ to indicate direction 
make one direction in space expressible by real numbers, the re- 
mainder being imaginaries of some kind, leading in general to 
equations which are heterogeneous ; Hamilton makes all directions 
in space equally imaginary, or rather equally real, thereby ensuring 
to his Calculus the power' of dealing with space indifferently in 
all directions. 

In fact, as we shall see, the Quaternion method is independent 
of axes or any supposed directions in space, and takes its reference 
lines solely from the problem it is applied to. 

14.] But, for the purpose of elementary exposition, it is best 
to begin by assimilating it as closely as we can to the ordinary 
Cartesian methods of Geometry of Three Dimensions, which are 
in fact a mere particular case of Quaternions in which most of 
the distinctive features are lost. We shall find in a little that 
it is capable of soaring above these entirely, after having employed 
them in its establishment; and, indeed, as the inventor's works 
amply prove, it can be established, ah initio^ in various ways, 
without even an allusion to Cartesian Geometry. As this work 
is written for students acquainted with at least the elements of 
the Cartesian method, we keep to the first-mentioned course of 
exposition ; especially as we thereby avoid some reasoning which, 
though rigorous and beautiful, might be apt, from its subtlety, 
to prove repulsive to the beginner. 

We commence, therefore, with some very elementary geometrical 

15.] Suppose we have two points A and B in space, and suppose 
A given, on how many numbers does ^'s relative position depend ? 

If we refer to Cartesian coordinates (rectangular or not) we find 


that the data required are the excesses of ^'s three coordinates over 

those of A. Hence three numbers are required. 

Or we may take polar coordinates. To define the moon's position 
with respect to the earth we must have its Geocentric Latitude 
and Longitude, or its Right Ascension and Declination, and, in 
addition, its distance or radius-vector. Three again. 

16.J Here it is to be carefully noticed that nothing has been 
said of the actual coordinates of either A or B, or of the earth 
and moon, in space ; it is only the relative coordinates that are 

Hence any expression, as AB, denoting a line considered with 
reference to direction as well as length, contains implicitly three 
numbers, and all lines parallel and equal to AB depend in the same 
way upon the same three. Hence, all lines which are equal and 
parallel may he represented hy a common symbol, and that symbol 
contains three distinct numbers. In this sense a line is called a 
VEOTOE, since by it we pass from the one extremity, A, to the 
other, B ; and it may thus be considered as an instrument which 
carries A to B : so that a vector may be employed to indicate a 
definite translation in space. 

17.] We may here remark, once for all, that in establishing a 
new Calculus, we are at liberty to give any definitions whatever 
of our symbols, provided that no two. of these interfere with, or 
contradict, each other, and in doing so in Quaternions simplicity 
and (so to speak) naturalness were the inventor's aim. 

18.] Let AB be represented by a, we know that a depends on 
three separate numbers. Now if CD be equal in length to AB 
and if these lines be parallel, we have evidently CI) = AB = a, 
where it will be seen that the sign of equality, between vectors 
contains implicitly equality in length and parallelism in direction. 
So far we have extended the meaning of an algebraical symbol. 
And it is to be noticed that an equation between vectors, as 

a = /3, 
contains three distinct equations between mere numbers. 

19.] We must now define + (and the meaning of — will follow) 
in the new Calculus. Let A, B, C be any three points, and (with 
the above meaning of =) let 

AB=a, BG=I3, AC=y. 

If we define + (in accordance with the idea (§ 16) that a vector 
represents a translation) by the equation 


or AB + £C = AC, 

we contradict nothing that precedes, but we at once introduce the 
idea that vectors areata he compounded, in direction and magnitude, 
like simultaneous velocities. A reason for this may be seen in 
another way if we remember that by adding the diflferences of the 
Cartesian coordinates of A and B, to those of the coordinates of 
B and C, we get those of the coordinates of A and C. Hence these 
coordinates enter" linearly into the expression for a vector. 

20.] But we also see that if C and A coincide (and C may be 
any point) AQ = 0, 

for no vector is then required to carry A to C. Hence the above 
relation may be written, in this case, 

AB+BA = 0, 
or, introducing, and by the same act defining, the symbol — , 

Hence, t/ie symbol — , applied to a vector, simply shows that its 
direction is to he reversed. 

And this is consistent with all that precedes ; for instance, 

ab+bc = m;, 

and AB = AC-BC, 

or =AG+CB, 

are evidently but different expressions of the same truth. 
21.] In any triangle, ABC, we have, of course, 

IB + BC+CA^ 0; 
and, in any closed polygon, whether plane or gauche, 

AB-k^BC+ '. + TZ+ZA = 0. 

In the case of the polygon we have also 

AB + BC+ + fZ=AZ. 

These are the well-known propositions regarding composition of 
velocities, which, by the second law of motion, give us the geo- 
metrical laws of composition of forces. 

22.] If we compound any number of parallel vectors, the result 
is obviously a numerical multiple of any one of them. 
Thus, if A, B, C are in one straight line, 

where a; is a number, positive when B lies between A and C, other- 
wise negative : but such that its numerical value, independent 
of sign, is the ratio of the length of BC to that of AB. This is 


at oace evident if AB and BC be commensurable j and is easily 
extended to incommensurables by the usual reductio ad absurdum. 

23.] An important, but almost obvious, proposition is that any 
vector may he resolved, and in one way only, into three components 
parallel respectively to any three given vectors, no two of which are 
parallel, and which are not parallel to one plane. 

Let OA, OB, OC be the three fixed vectors, c 
OP any other vector. From P draw PQ 
parallel to CO, meeting the plane BOA in Q. 
[There must be a definite point Q, else PQ, 
and therefore CO, would be parallel to BOA, 
a case specially excepted.] Prom Q draw 
QB parallel to BO, meeting OA in B. Then 
we have OP = 0^ + ^ + QP (§ 21), 
and these components are respectively parallel to the three given 
vectors. By § 22 we may express OB as a numerical multiple 
of OA, RQ oi OB, and QP of OC. Hence we have, generally, for 
any vector in terms of three fixed non-coplanar vectors, a, /3, y, 

OP = p = xa + yl3 + zy, 
which exhibits, in one form, the three numbers on which a vector 
depends (§ 16). Here x, y, z are perfectly definite, and can have 
but single values. 

24.] Similarly any vector, as OQ, in the same plane with OA 
and OB, can be resolved into components OB, RQ, parallel re- 
spectively to OA and OB ; so long, at least, as these two vectors 
are not parallel to each other. 

25.] There is particular advantage, in certain cases, in employ- 
ing a series of three mutually perpendicular unit-vectors as lines of 
reference. This system Hamilton denotes by i,j, h. 

Any other vector is then expressible as 
p = xi-\-yj-\-zh. 
Since i, j, k are unit- vectors, x, y, z are here the lengths of con- 
terminous edges of a rectangular parallelepiped of which p is the 
vector-diagonal ; so that the length of p is, in this case, 

Let TO- = ^i + T/y+C^ 

be any other vector, then (by the proposition of § 23) the vector 
equation p =. 'ss 

obviously involves the following three equations among numbers, 
« = ^. y = ■<), z=C 


Suppose i to be drawn eastwards, J northwards, and k upwards, 
this is equivalent merely to saying that if two points coincide, ihey 
are equally to tie east {or west) of any third point, equally to the 
north {or south) of it, and equally elevated above {or depressed below) 
its level. 

26.] It is to be carefully noticed that it is only when a, fi, y are 
not coplanar that a vector equation such as 

p = OT, 

or «a-f ^;3 + «y = £o + jj/3+Cy, 

necessitates the three numerical equations 

m = i, y = n, « = ^ 
For, if a, ^j y be coplanar (§ 24), a condition of the following form 
must hold y = aa + b^. 

Hence p = {x + za)a+{y + zh)p, 

^={i+Ca)a + {r, + Cb)^, 
and the equation p ■= m 

now requires only the two numerical conditions 

x + za= ^+Ca, y + zb = r] + (b. 

27.] The Commutative and Associative Laws hold in the combination 
of vectors by the signs + and — . It is obvious that, if we prove 
this for the sign + , it will be equally proved for — , because — 
before a vector (§ 20) merely indicates that it is to be reversed 
before being considered positive. 

Let A, B, G, B be, in order, the corners of a parallelogram ; we 
have, obviously, Jb = SC, IT) = BG. 

And AB + BC= IC=An+BC=BC+AB. 

Hence the commutative law is true for the addition of any two 

vectors, and is therefore generally true. 

Again, whatever four points are represented by A, B, C, J), we 
have lD = IB+BB = AC-\-UD, 

or substituting their values for AB, BB, AC respectively, in these 
three expressions, 

lB+BC-\-CB^AB + {BC+CB)= {AB + BC) + CB. 
And thus the truth of the associative law is evident. 

28.] The equation „ — xB, 

where p is the vector connecting a variable point with the origin, 
/3 a definite vector, and x an indefinite number, represents the 
straight line drawn from the origin parallel to ^ (§ 22). 


The straight line drawn from A, where OA = a, and parallel 
to j8, has the equation 

p = a + a;/3 ; (1) 

In words, we may pass directly from to P by the vector OP or p ; 
or we may pass first to A, by means of OA or a, and then to P 
along a vector parallel to ^ (§ 16). 

Equation (1) is one of the many useful forms into which Quater- 
nions enable us to throw the general equation of a straight line in 
space. As we have seen (§ 25) it is equivalent to three numerical 
equations ; butj as these involve the indefinite quantity x, they are 
virtually equivalent to but two, as in ordinary Geometry of Three 

29.] A good illustration of this remark is furnished by the fact 
that the equation p = va + se^, 

which contains two indefinite quantities^ is virtually equivalent to 
only one numerical equation. And it is easy to see that it re- 
presents the plane in which the lines o and fi lie ; or the surface 
which is formed by drawing, through every point of OA, a line 
parallel to OB. In fact, the equation, as written, is simply § 24 
in symbols. 

And it is evident that the equation 

P = y+ya + oo^ 
is the equation of a plane passing through the extremity of y, and 
parallel to a and ;8. 

It will now be obvious to the reader that the equation 

P =i'i«i+i'2a2+ = '2pa, 

where a^, Og, &c. are given vectors, and Pi,P2> ^c. numerical quan- 
tities, represents a strd/igM line if i5i,j»2J &c. be linear functions of 
one indeterminate number ; and a plane, if they be linear expres- 
sions containing two indeterminate numbers. Later (§31 {})), this 
theorem will be much extended. 

Again, the equation p = xa + y^-^zy 
refers to any point whatever in space, provided a, /3, y are not 
coplanar. {Ante, § 23). 

30.] The equation of the line joining any two points A and B, 
where OA = a and OB = 13, is obviously 
P = a + a;(/3— a), 
or p = ^ + y(a-/3). 

These equations are of course identical, as may be seen by putting 
1—y for ss. 

10 QUATERNIONS. [3 1. 

The first may be written 

p + (x—l)a — x^ = ; 
or j)p + qa + rfi = 0, 
subject to the condition p + q + r = identically. That is — A 
homogeneous linear function of three vectors, equated to zero, 
expresses that the extremities of these vectors are in one straight 
line, if the sum of the coefficients he identically zero. 

Similarly, the equation of the plane containing the extremities 
A, B, C of the three non-coplanar vectors a, j3, y is 

p = a + a:(/3-a) + y(y-/3), 
where x and y are each indeterminate. 

This may be written 

pp + qa + r^ + sy = 0, 
with the identical relation 

p + q + r + s = 0. 
which is the condition that four points may lie in one plane. 

31.] We have already the means of proving, in a very simple 
manner, numerous classes of propositions in plane and solid geo- 
metry. A very few examples, however, must suflSce at this stage ; 
since we have hardly, as yet, crossed the threshold of the subject, 
and are dealing with mere linear equations connecting two or more 
vectors, and even with them we are restricted as yet to operations of 
mere addition. We will give these examples with a painful minute- 
ness of detail, which the reader will soon find to be necessary only 
for a short time, if at all. 

(a.) The diagonals of a parallelogram bisect each other. 
Let ABGB be the parallelogram, the point of intersection of 
its diagonals. Then 

iO + ^= IB =^G = Bb+OC, 
which gives AO-OC = BO-OB. 

The two vectors here equated are parallel to the diagonals respect- 
ively. Such an equation is, of course, absurd unless 

(1) The diagonals are parallel, in which case the figure 

is not a parallelogram ; 

(2) Jo = Oa, and ^ = OB, the proposition. 

(i.) To show that a triangle can he constructed, whose sides 
are parallel, and equal, to the hisectors of the sides of 
any triangle. 
Let ABC be any triangle, Aa, Bh, Cc the bisectors of the sides. 



Then Aa, = AB+Ba = AB^-\ BC, 

Bb - - - =BC+hCA, 

Co - - - =Cl+\lB. 

Hence Aa + Bb + Cc=^{lB + BG+CA)^(); 

which (§21) proves the proposition. 
Also Aa — JS+\BC 

= AB-\{Cl+AB) 
= ^{AB-ai) = i{lB+IC), 
results which are sometimes useful. They may be easily verified 
by producing Aa to twice its length and joining the extremity 
with B. 

{¥.) The bisectors of the sides of a triangle meet in a point, which 
trisects each of them. 
Taking A as origin, and putting o, /3, y for vectors parallel, and 
equal, to the sides taken in order BC, CA, AB; the equation of 
Bb is (§ 28 (1)) 

p = y + «(y+f) = (i+a')y + |^- 

That of Cc is, in the same way, 

p = -(l + y)^-|y. 
At the point 0, where Bb and Cc intersect, 

Since y and ^ are not parallel, this equation gives 
H-«' = -f, and | = _(i+y). 

From these a; = y = — |. 

Hence iO = 4 (y-/3) = 1 2a. (See Ex. («).) 
This equation shows, being a vector one, that Aa passes through 0, 
and that JO : Oa : : 2: 1. 

(c) If 02 = a, ^ 

be three given co-planar 



^^"^^^ / 



vectors, and the lines in- 



dicated in the figure be drawn, the points «i,*i,Ci lie in a straight 




We see at once, by the process indicated in § 30, that 

Oc = 

aa + b^ Qj ^ "-"■ 

a + b 
Hence we easily find 

-a — 2b ^ 

0-a= '^ 


Oa^ =■ 

Oc, = — 7 

^ b—a 

\—a-2b •■ l — 2a-b' 

These give 

-{l-a-2b)Oai+{l-2a-6)Obi-{b-a)Oc^ = 0. 
But ■ _(l-a-2i) + (l-2a-5)-(5-a) = identically. 
This, by § 30, proves the proposition. 

(d.) Let 02 = a, OB = /3, be any two vectors. If MP be 

parallel to OB; and OQ, BQ, be drawn parallel to AP, 

OP respectively; the locus of Q is a straight line parallel 

to OA. 

Let OM = ea. 

AP = e— la + a;)3. 
Hence the equation of 

p = y(e-la+»i3); 
and that of ^Q is 

p = ^ + z{ea+co^). 

At Q we have, therefore, 

xy = l+zx, \ 

y{e-\) = ze.\ 
These give xy = e, and the equation of the locus of Q is 

P = «/3 + /a, 
i. e. a straight line parallel to OA, drawn through N in OB pro- 
duced, so that ON -.OB:: OM: OA. 

CoE. li BQ meet MP in q,P'c[=^; and if AP meet NQ in p, 
Qp=a. _ _ 

Also, for the point B we have pB = AP, Q,R=Bq. 

Hence, if from any two points, A and B, lines be drawn intercepting 
a given length Pq on a given line Mq; and if, from B their point of 
intersection, Bp be laid off = PA, and BQ = qB ; Q and p lie on a 
fixed straight line, and the length of Qp is constant. 

(e.) To find the centre of inertia of any system. 

If OA = a, OB = a^, be the vector sides of any triangle, the 
vector from the vertex dividing the base AB in C so that 




BG : CA ■.:m:m-^ is 

For A£ is Oj — a, and therefore AC is 

-(oj— a). 



OA + AC 

= a + 

(«! — a) 

.,i + Ml 

ma + mj^a^ 

This expression shows how to find the centre of inertia of two 
masses; m at the extremity of a, m^ at that of a^. Introduce Wg 
at the extremity of 02, then the vector of the centre of inertia of the 
three is, by a second application of the formula, 


M a + »Zi Oj^ + »«2 02 


(m+mj^ + m^ 

For any number of masses, expressed generally by m at the extre- 
mity of the vector a, we have the vector of the centre of inertia 

'^ s(ot) ■ 

This may be written 2m(a—fi) = 0. 

Now Oj— /3 is the vector of % with respect to the centre of inertia. 
Hence the theorem, ^ the vector of each element of a mass, drawn 
from the centre of inertia, he increased in length in proportion to the 
mass of the element, the sum of all these vectors is zero. 

(_/.) We see at once that 
the equation 

where t is an indeterminate 
number, and a, j8 given vec- 
tors, represents a parabola. 
The origin, 0, is a point on 
the curve, /3 is parallel to 
the axis, i. e. is the diameter 
OB drawn from the origin, 
and a is OA the tangent at the origin. 

qp = at, 6q = 


The secant joining the points where t has the values t and If is 
represented by the equation 

, = a. + ^+.(ar+^-a.-^) (§30) 

Put 1f=.t, and write x for x{if—t) [which may have any value] 
and the equation of the tangent at the point {t) is 

Put X = —t, p = — > 

or the intercept of the tangent on the diameter is —the abscissa of 
the point of contact. 

Otherwise: the tangent is parallel to the vector a+fit or 

at + pt'' OT at + if. + if. ov 0Q+ UP. BuifF=fd + 6P, 
hence TO = OQ. 
•{ff.) Since the equation of any tangent to the parabola is 

p = at + ^ + x{a+lii), 

let us find the tangents which can be drawn from a given point. 
Let the vector of the point be 

p=pa + ql3 (§24). 
Since the tangent is to pass through this point, we have, as con- 
ditions to determine t and x, i + x = p, 

-j + xt = q; 

by equating respectively the coefficients of a and /3. 

Hence ^ =jo+ \/^^ — 2$'. 

ThuSj in general, two tangents can be drawn from a given point. 

These coincide if ^2 _ 2^ ; 

that is, if the vector of the point from which they are to be drawn 

is „ P^ „ 

p =pa + qfi =Pa.+ ^p, 

i. e. if the point lies on the parabola. They are imaginary if 
2q>p^, i. e. if the point be 

r being positive. Such a point is evidently within the curve, as at 
72, where OQ =^/3, QP=pa, PB = r^. 


(A.) Calling the values o{ t f9r the two tangents found in (^) 
ti and ^2 respectively, it is obvious that the vector join- 
ing the points of contact is 



which is parallel to f ^^ 

a + ^-i^; 

or, by the values of t^ and t^ in (ff), 

Its direction, therefore, does not depend on q. In words, If pairs 
of tangents he ckamn to a parabola from points of a diameter pro&uced, 
the chords of contact are parallel to the tangent at the vertex of the 
diameter. This is also proved by a former result, for we must have 
OT for each tangent equal to Q 0. 

{i.) The equation of the chord of contact, for the point whose 

vector is p=pa + ql3, 

Bt ^ 
is thus P = a^i+^ + ^(«+J0;8). 

Suppose this to pass always through the point whose vector is 

p = aa + b^. 

Then we must have , , 

h+^ = «. ) 
t ^ [ 

or ti=p±^p^ — 2pa + 2b. 

Comparing this with the expression in {g), we have 

q =pa—h; 
that is, the point from which the tangents are drawn has the vector 
p =pa + {pa—b)^ 
= —b^+p{a + aj3), a straight line (§ 28 (1)). 
The mere form of this expression contains the proof of the usual 
properties of the pole and polar in the parabola ; but, for the sake 
of the beginner, we adopt a simpler, though equally general, 

Suppose a = 0. This merely restricts the pole to the particular 
diameter to which we have referred the parabola. Then the pole 
is Q, where p = b^; 

and the polar is the line TU, for which 

p = -6fi+pa. 


Hence the polar of any point is parallel to the tangent at the extremity 
of the diameter on which the point lies, and its intersection with that 
diameter is as far beyond the vertex as the pole is within, and vice 

(J.) As another example let us prove the following theorem. 

Jf a triangle he inscribed in a parabola, tlie three points 

in which the sides are met by tangents at the angles lie in 

a straight line. 

Since is any point of the curve, we may take it as one corner 

of the triangle. Let t and ^j determine the others. Then, if 

OTj, OTj, iirg represent the vectors of the points of intersection of the 

tangents with the sides, we easily find 


''' = t:vt"" 

These values give 

itj^-t 2t—t-i^ t^-t^ 

Ai "^h-t ^t-h ty-^ „ •. ^. 1, 
Also — i — -i- — i- — = identically. 

Hencoj by § 30, the proposition is proved. 

ijc) Other interesting examples of this method of treating 
curves will, of course, suggest themselves to the 
student. Thus 

p = a cos if 4- ^ sin ^ 
or p = oa;+^^/l— jc^ 

represents an ellipse, of which the given vectors a and /3 are semi- 
conjugate diameters. 

Agam, p = aif + - or p = a tana;+^cota; 

evidently represents a hyperbola referred to its asymptotes. 

But, so far as we have yet gone with the explanation of the 
calculus, as we are not prepared to determine the lengths or in- 
clinations of vectors, we can investigate only a very small class of 
the properties of curves, represented by such equations as those 
above written. 


(I.) We may now, in extensi^ of the statement in § 29, make 
the obvious remark that 
p = Sj)a 
is the equation of a curve in space, if the numbers i'i,i»2> ^^- ^^^ 
functions of one indeterminate. In such a case the equation is 
sometimes written . _ j^/f. 

But, if jOi, j»2J ^c. be functions of two indeterminates, the locus of 
the extremity of p is a surface ; whose equation is sometimes written 

p = <t>{t,u). 
(m.) Thus the equation 

p = acost + ^sint+yt 
belongs to a helix. 

Again, p=pa + q^ + ry 

with a condition of the form 

ap^ 4 6q^ + cr^ ■= I 
belongs to a central surface of the second order, of which a, p, y 
are the directions of conjugate diajneters. If a, b, c be all positive, 
the surface is an ellipsoid. 

32.] In Example (_/) above we performed an operation equi- 
valent to the differentiation of a vector with reference to a single 
numerical variable of which it was given as an explicit function. 
"As this process is of very great use, especially in quaternion investi- 
gations connected with the motion of a particle or point ; and as it 
will afford us an opportunity of making a preliminary step towards 
overcoming the novel difficulties which arise in quaternion differen- 
tiation; we will devote a few sections to a more careful exposition 
of it. 

33.] It is a striking circumstance, when we consider the way 
in which Newton's original methods in the Differential Calculus 
have been decried, to find that Hamilton was obliged to employ 
them, and not the more modern forms, in order to overcome the 
characteristic difficulties of quaternion differentiation. Such a thing 
as a differential coefficient has absolutely no meaning in quaternions, 
except in those special cases in which we are dealing with degraded 
quaternions, such as nvmibers, Cartesian coordinates, &c. But a 
quaternion expression has always a differential, which is, simply, 
what Newton called sifluadon. 

As with the Laws of Motion, the basis of Dynamics, so with the 
foundations of the Differential Calculus ; we are gradually coming 
to the conclusion that Newton's system is the best after all. 


18 QtTATERNIONS. [34- 

34.J Suppose p to be the vector of a curve in space. Then, 
generally, p may be expressed as the sum of a number of terms, 
each of which is a multiple of a given vector by a function of some 
one indeterminate; or, as in § 31 (1), if P be a point on the curve, 
6P=p = 4>{t). 

And, similarly, if Q be ani/ other point on the curve, 

where htis any number whatever. 

The vector-chord PQis therefore, rigorously, 
6p = pi-p = (f>{t + bt)-cl>t. 

35.] It is obvious that, in the present case, because the vectors 
involved in (j) are constant, and their numerical multipliers alone vary, 
the expression i^it^ht) is, by Taylor's Theorem equivalent to 


Hence, ^<^(0 ,,^^'<^ W (8^)' ^^ 

And we are thus entitled to write, when ht has been made inde- 
finitely small, , , ,,. 

^ ' ,lp. dp dc\,{t) _ , 

In such a case as this, then, we are permitted to differentiate, 
or to form the differential coefficient of, a vector, according to the 
ordinary rules of the Differential Calculus. But great additional 
insight into the process is gained by applying Newton's method. 

36.] Let OP be 
_ P = <i>[t), 
and OQi 

p= 4>{t + dt), 
where dt is any number whatever. 

The number t may here be taken 
as representing time, i. e. we may 
suppose a point to move along the 
curve in such a way that the value 
of t for the vector of point P of the 
curve denotes the interval which has 
elapsed (since a fixed epoch) when the moving point has reached 
the extremity of that vector. If, then, dt represent any interval, 
finite or not, we see that 

will be the vector of the point after the additional interval dt. 


But this, in general, gives us little or no information as to the 
velocity of the point at P. We shall get a better approximation 
by halving the interval di, and finding Q^, where 0^2= <^ {i+h ^i)) 
as the position of the moving point at that time. Here the vector 
virtually described in ^df is PQ^. To find, on this supposition, 
the vector described in di, we must double PQ2) and we find, as a 
second approximation to the vector which the moving point would 
have described in time dt, if it had moved for that period in the 
direction and with the velocity it had at P, 
Tq2=2PQ^ = 2{0Q2-6P) 

= 2{(l>{i+kdt)-(l){t)}. 
The next approximation gives 

P^, = 3PQ,= 3{6Q,-6P) 

= 3{4>{i+idt)-<t>{i)]. 
And so on, each step evidently leading us nearer the sought truth. 
Hence, to find the vector which would have been described in time 
dt had the circumstances of the motion at P remained undisturbed, 
we must find the value of 

dp = Tq = J^:,=^ai^cj>(t + ^dt)-<j>{t)\- 

We have seen that in this particular case we may use Taylor's 
Theorem. We have, therefore, 

dp = J^,=^ X \^'{t)\ di+<j>"{t) ^ ^ 4 &c. I 

= 4)' (t) dt. 
And, if we choose, we may now write 

37.] But it is to be most particularly remarked that in the 
whole of this investigation no regard whatever has been paid to 
the magnitude of dt. The question which we have now answered 
may be put in the form — A point describes a given curve in a given 
manner. At any point of its path its motion suddenly ceases to he 
accelerated. What space will it describe in a definite interval ? As 
Hamilton well observes, this is, for a planet or comet, the case 
of a ' celestial Atwood's machine.' 

38.] If we suppose the variable, in terms of which p is expressed, 

to be the arc, s, of the curve measured from some fixed point, we 

find as before , ,,,,, ,, ^'{t)ds 

dp = ^{t)dt = ^^-^ 

= 4>'{s}ds. ^^ 

C 3 


From the very nature of the question it is obvious that the length 
of dp must in this case 'be ds. This remark is of importance, as 
we shall see later ; and it may therefore be useful to obtain afresh 
the above result without any reference to tiine or velocity. 

39.] Following strictly the process of Newton's Vllth Lemma, 
let us describe on Pq^ an arc similar to PQg, and so on. Then 
obviously, as the subdivision of ds is carried farther, the new arc 
(whose length is always ds) more and more nearly coincides with 
the line which expresses the corresponding approximation to <?p. 

40.] As a final example let us take the hyperbola 

Here dp = {a-^)dt. 

This shews that the tangent is parallel to the vector 

at --• 

In words, if the vector {from the centre) of a point in a hyperbola 
he one diagonal of a parallelogram, two of whose sides coincide with 
the asymptotes, the other diagonal is parallel to the tangent at the 

41.] Let us reverse this question, and seek the envelope of a line 
which cuts off from two fixed axes a triangle of constant area. 

If the axes be in the directions of a and fi, the intercepts may 

evidently be written at and y . Hence the equation of the line is 

p = at-\-x{Y—aty 

The condition of envelopment is, obviously, (see Chap. IX.) 

dp = 0. 

This gives =\a-x{^ + a)\dt+ {^-at)dx*. 

Hence {\—x)dt — tdx=0, 

J X ^, dx ^ 

and — — dt+ -^ = 0. 

* We are not here to equate to zero the coefficients of dt and dx; for we must 
remember that this equation is of the form 

=pa + q$, 

where p and q are numbers ; and that, so long as a and are actual and non-parallel 
vectors, the existence of such an equation requires 

i> = 0, 5 = 0. 


From these, at once, x = \, since dx and dt are indeterminate. 
Thus the equation of the envelope is 

the hyperbola as before ; a, ;3 being portions of its asymptotes. 

42.] It may assist the student to a thorough comprehension 
of the above process, if we put it in a slightly different form. 
Thus the equation of the enveloping line may be written 

p = ai!(l-a!) + /3*, 

which gives dp = = ad {t (1 —x))+^d (-) • 

Hence, as a is not parallel to /3, we must have 

d{t{l-x)) = (i, ^(f) = 0; 

and these are, when expanded, the equations we obtained in the 
preceding section. 

43.] For farther illustration we give a solution not directly em- 
ploying the differential calculus. The equations of any two of the 
enveloping lines are 

p = at + X 0-: at\t 

p =aifi + a?i(— -ai5i)> 

t and <i being given, while x and ajj are indeterminate. 
At the point of intersection of these lines we have (§ 26), 

t{l-x) = ^i(l-«i), \ 
X _Xi > 

These give, by eliminating x^, 

t{\-x) = ti{l-^x), 


or X =■ . • 

ti + t 

Hence the vector of the point of intersection is 


and thus, for the ultimate intersections, where ^^ = 1, 
p = ^ (a^ + y) as before. 

CoE. (1). If tt^ = 1, 

a + /3 . 

or the intersection lies in the diagonal of the parallelogram on a, j3. 
Cob. (2). If ti = mt, where m is constant, 

mta + — 


P = ■ 

But we have also iv = 

m+ 1 


Hence tAe locus of a point which divides in a given ratio a line 
cutting off a given area from, two fixed axes, is a hyperbola of which 
these axes are the asymptotes. 

Cor. (3). If we take 

tt^ (^+ ^1) = constant 
the locus is a parabola ; and so on. 

44.] The reader who is fond of Anharmonic Ratios and Trans- 
versals will find in the early chapters of Hamilton's Elements of 
Quaternions an admirable application of the composition of vectors 
to these subjects. The Theory of Geometrical Nets, in a plane, 
and in space, is there very fully developed ; and the method is 
shewn to include, as particular cases, the processes of Grassmann's 
Ausdehnungslehre and Mobius' Barycentrische Calcul. Some very 
curious investigations connected with curves and surfaces of the 
second and third orders are also there founded upon the composition 
of vectors. 


1. The lines which join, towards the same parts, the extremities 
of two equal and parallel lines are themselves equal and parallel. 
{Euclid, I. xxxiii.) 

2. Find the vector of the middle point of the line which joins 


the middle poiats of the diagonals of any quadrilateral, plane or 
gauche^ the vectors of the corners being given ; and so prove that 
this point is the mean point of the quadrilateral. 

If two opposite sides be divided proportionally, and two new 
quadrilaterals be formed by joining the points of division, the mean 
points of the three quadrilaterals lie in a straight line. 

Shew that the mean point may also be found by bisecting the 
line joining the middle points of a pair of opposite sides. 

3. Verify that the property of the coefficients of three vectors 
whose extremities are in a line (§ 30) is not interfered with by 
altering the origin. 

4. If two triangles ABC, abc, be so situated in space that Aa, 
Bb, Cc meet in a point, the intersections of AB, ah, of BG, be, and 
of CA, ca, lie in a straight line. 

5. Prove the converse of 4, i. e. if lines be drawn, one in each 
of two planes, from any three points in the straight line in which 
these planes meet, the two triangles thus formed are sections of 
a common pyramid. 

6. If five quadrilaterals be formed by omitting in succession each 
of the sides of any pentagon, the lines bisecting the diagonals of 
these quadrilaterals meet in a point. (H. Fox Talbot.) 

7. Assuming, as. in § 7, that the operator 

cos 6 + \/— 1 sin 6 
turns any radius of a given circle through an angle 6 in the 
positive direction of rotation, without altering its length, deduce 
the ordinary formulae for cos [A + B), cos {A—B), sin {A + B), and 
sin [A—B), in terms of sines and cosines of A and B. 

8. If two tangents be drawn to a hyperbola, the line joining 
the centre with their point of intersection bisects the lines joining 
the points where the tangents meet the asymptotes : and the 
tangent at the point where it meets the curves bisects the intercepts 
of the asymptotes. 

9. Any two tangents, limited by the asymptotes, divide each 
other proportionally. 

10. If a chord of a hyperbola be one diagonal of a parallelogram 
whose sides are parallel to the asymptotes, the other diagonal passes 
through the centre. 

11. Shewthat p = x^ a + f ^-\-{x-\-yf y 

is the equation of a cone of the second degree, and that its section 
by the plane _ pa + g^ + ry 

^~ p + q+r 


is an ellipse which touches, at their middle points, the sides of 
the triangle of whose corners a, /3, y are the vectors. (Hamilton, 
Elements, p. 96.) 

12. The lines which divide, proportionally, the pairs of opposite 
sides of a gauche quadrilateral, are the generating lines of a hyper- 
bolic paraboloid. (Ibid. p. 97.) 

13. Shew that p = x^a + y^fi + z^y, 
where x + y + z = 0, 

represents a cone of the third order, and that its section by the plane 

'' ~ p + q + r 
is a cubic curve, of which the lines 

P + 1 
are the asymptotes and the three (real) tangents of inflexion. Also 
that the mean point of the triangle formed by these lines is a 
conjugate point of the curve. Hence that the vector a-f-(3 + y is a 
conjugate ray of the cone. (Ibid. p. 96.) 



45.] We now come to the consideration of points in which the 
Calculus of Quaternions differs entirely from any previous mathe- 
matical method ; and here we shall get an idea of what a Qua- 
ternion is, and whence it derives its name. These points are 
fundamentally, involved in the novel use of the symbols of mul- 
tiplication and division. And the simplest introduction to the 
subject seems to be the consideration of the quotient, or ratio, of 
two vectors. 

46.] If the given vectors be parallel to each other, we have 
already seen (§ 22) that either may be expressed as a numerical 
multiple of the other; the multiplier being simply the ratio of 
their lengths, taken positively if they are similarly directed, nega- 
tively if they run opposite ways. 

47.] If they be not parallel, let OA and OB be drawn parallel 
and equal to them from any point ; and the question is reduced 
to finding the value of the ratio of two vectors drawn from the 
same point. Let us try to find upon how many distinct numbers this 
ratio depends. 

We may suppose OA to be changed into 0£ by the following 

1st. Increase or diminish the length of OA till it becomes 
equal to that of OB. For this only one number is 
required, viz. the ratio of the lengths of the two 
vectors. As Hamilton remarks, this is a positive, or 
rather a signless, number. 
2nd. Turn OA about until its direction coincides with that 
of OB, and (remembering the effect of the first operation) 


we see that the two vectors now coincide or become 
identical. To specify this operation three more numbers 
are required, viz. two angles (such as node and inclina- 
tion in the case of a planeVs orbit) to fix the plane in 
which the rotation takes place, and one angle for the 
amount of this rotation. 
Thus it appears that the ratio of two vectors, or the multiplier 
required to change one vector into another, in general depends upon 
four distinct numbers, whence the name quaternion. 

The particular case of perpendicularity of the two vectors, where 
their quotient is a vector perpendicular to their plane, is fully con- 
sidered below ; §§ 64, 65, 72, &c. 

48.] It is obvious that the operations just described may be 
performed, with the same result, in the opposite order, being per- 
fectly independent of each other. Thus it appears that a quaternion, 
considered as the factor or agent which changes one definite vector 
into another, may itself be decompofed into two factors of which 
the order is immaterial. 

The stretching factor, or that which performs the first operation 
in § 47, is called the Tensou, and is denoted by prefixing T to the 
quaternion considered. 

The turninff factor, or that corresponding to the second operation 
in § 47, is called the Versoe, and is denoted by the letter U prefixed 
to the quaternion. 

49.] Thus, if OA = a, OB = fi, and if q be the quaternion which 
changes a to /3, we have 

13 = qa, 

which we may write in the form 

— = q, or ^a-i = q, 

if we agree to defne that 

— .a = |3a-i. o = p. 

Here it is to be particularly noticed that we write q before a to 
signify that a is multiplied by q, not q multiplied by a. 

This remark is of extreme importance in quaternions, for, as we 
shall soon see, the Commutative Law does not generally apply to 
the factors of a product. 

We have also, by §§ 47, 48, 



where, as before, Tq^ depends merely on the relative lengths of 
a and j3, and Uq depends solely on their directions. 

Thus, if Oi and jSj be vectors of unit length parallel to a and j3 
respectively, ^^^^^ u^^- = U^. 

As will soon be shewn, when a is perpendicular to ^, the versor of 
the quotient is quadrantal, i. e. it is a unit-vector. 

50.] We must now carefully notice that the quaternion which 
is the quotient when /3 is divided by a in no way depends upon 
i\^e- absolute lengths, or directions, of these vectors. Its value 
will remain unchanged if we substitute for them any other pair 
of vectors which 

(1) have their lengths in the same ratio, 

(2) have their common plane the same or parallel, 
and (3) make the same angle with each other. 

Thus in the annexed figure 

6^1^ OB 

0^1 " OA 
if, and only if, 

^^^ O.Aj^ OA 

(2) plane AOS parallel to plane A^O^B^, 

(3) I.AOB = LA^O^B^. 
[Equality of angles is understood to include 

similarity in direction. Thus the rotation about 
an upward axis is negative (or right-handed) 
from OA to OB, and also from Oj A^ to 0^ B^r\ 

51.1 The Reciprocal of a quaternion q is defined by the equation, 

1 -1 1 

q- = qi ^=1. 

Hence if - = ?> ov 



a 1 J 

we must have 'B~a ~ ^ ' 

a ■, 

For this gives -.p = q '^.qa, 

and each member of the equation is evidently equal to a. 



Or, we may reason thus, q changes 61 to 0£, q-^ must therefore 
change OB to OA, and is therefore expressed by - (§ 49). 

The tensor of the reciprocal of a quaternion is therefore the 
reciprocal of the tensor ; and the versor differs merely by the reversal 
of its representative angle. The versor, it must be remembered, 
gives the plane and angle of the turning — ^it has nothing to do 
with the extension. 

52.] The Conjugate of a quaternion q, written Kq, has the same 
tensor, plane, and angle, only the angle is taken the reverse way. 
Thus, if OA, OB, OA', lie in one plane, and if 

0A'= OA, and LA:0B=IA0B, we have 

OB ,6b ■ . n TT 

-z=^ = a, and-^=- = coniugate 01 q ■=■ Kq. 
OA ^' OA' ^^ ^ ^ 

By last section we see that 

Kq = {Tqfq-\ 
Hence qKq = Kq.q = {Tqf. 

This proposition is obvious, if we recollect that the 

" tensors of q and Kq are equal, and -that the versors 

are such that either annuls the effect of the other. The joint effect 

of these factors is therefore merely to multiply twice over by the 

common tensor. 

53.] It is evident from the results of § 50 that, if a and ^ be 
of equal length, their quaternion quotient becomes a versor (the 
tensor being unity) and may be represented indifferently by any 
one of an infinite number of ares of given length lying on the 
circumference of a circle, of which the two vectors are radii. This 
js of considerable importance in the proofs which follow. 

Thus the versor ^=^ may be represented 

in magnitude, plane, and direction (§ 50) 
by the arc AB, which may in this extended 
sense be written AB. 

And, similarly, the versor ' is repre- 

sented by A^B^ which is equal to (and 
measured in the same direction as) AB if 
jLAiOBi = LAOB, 
i.e. if the versors are equal, in the quaternion meaning of the word. 


54.] By the aid of this process,''Vhen a versor is represented as 
an arc of a great circle on the unit-sphere, we can easily prove that 
qiiaternion multiplication is not generally commutative. 

Thus let q be the versor j!b or ^=- • 
^ ^ OA 

Make BC = AB, (which, it must be 

remembered, makes the points A, B, C 

lie in one great circle), then q^ may also 

be represented by • 

^ OB 
In the same way any other versor r 

may be represented by BB or BB and bv -=- or -^=- • 

^ OB OB 

The line OB in the figure is definite, and is given by the inter- 
section of the planes of the two versors ; being the centre of the 

Now rOB = OB, and qOB = 00, 

Hence _ qrOB=6c, 

00 '-^ 

or qr = -=■ > and may therefore be represented by the arc BC of 

a great circle. 

But rq is easily seen to be represented by the arc AB. 

For q02 = OB, and rOB = OB, 

— OB 

whence rq OA = OB, and rq = -=- • 


Thus the versors rq and qr, though represented by arcs of equal 
length, are not generally in the same plane and are therefore un- 
equal : unless the planes of q and r coincide. 

Calling OA a, we see that we have assumed, or defined, in the 
above proof, that q.ra = qr.a and r.qa = r'q.a when qa, ra, q.ra, and 
r.qa are all vectors. 

55.J Obviously CB is Kq, BB is Kr, and CB is K(qr). But 
CB = BB.CB, which gives us the very important theorem 

K{qr) =Kr.Kq, 
i.e. the conjugate of tM product of two quaternions is the product of 
their conjugates in inverted order. 

56.] The propositions just proved are, of course, true of quater- 
nions as well as of versors ; for the former involve only an additional 


numerical factor which has reference to the length merely, and not 
the direction, of a vector (§48). 

57.] Seeing thus that the commutative law does not in general 
hold in the multiplication of quaternions, let us enquire whether 
the Associative Law holds. That is, itj), q, r be three quaternions, 
have we jtj.g'r = j)q.r ? 

This is, of course, obviously true if jo, q, r be numerical quantities, 
or even any of the imaginaries of algebra. But it cannot be con- 
sidered as a truism for symbols which do not in general give 

M = iP- 
58.] In the first place we remark that ^, q, and r may be con- 
sidered as versors only, and therefore represented by arcs of great 
. circles, for their tensors may obviously (§ 48) be divided out from 
both sides, being commutative with the versors. 
Let AB =p, BB = CA = q, aadi IE =z r. 
Join BC and produce the great circle till it meets EF in H, and 
make KH = FE = r, Bxidi EG = GB = pq (§ 54). 

Join GK. Then 
KG=HG.n]: = pq.r. 

Join FB and produce it to 
meet AB in M. Make 


and MN=AB, 
~^frlf^ and join NL. Then 

LN= MN.£M = p.qr. 

Hence to shew that p.qr = pq.r 
all that is requisite is to prove that LN, and KG, described as 
above, are equal a/rcs of the same great circle, since, by the figure, 
they are evidently measured in the same direction. This is perhaps 
most easily efiected by the help of the fundamental properties of 
the curves known as Sjokerical Conies. As they are not usually 
familiar to students, we make a slight digression for the purpose of 
proving these fundamental properties ; after Chasles, by whom and 
Magnus they were discovered. An independent proof of the asso- 
ciative principle will presently be indicated, and in Chapter VII 
we shall employ quaternions to give an independent proof of the 
theorems now to be established. 

59.*] Dbf. a spherical conic is the curve of intersection of a cone 
of the second degree with a sphere, the vertex of the cone being the 
centre of the sphere. 




Lemma. If a cone have one silHes of circular sections, it bas 
another series, and any two circles belonging to different series lie 
on a sphere. This is easily proved as follows. 

Describe a sphere, A, cutting the cone in one circular section, 
C, and in any other point whatever, and let the side OpP of the 
cone meet A in p, P; P being a point in C. Then PO-Op is 
constant, and, therefore, since P lies in a plane, p lies on a sphere, 
a, passing through 0. Hence the locus, c, of js is a circle, being 
the intersection of the two spheres A and a. 

Let OqQ be any other side of the cone, q and Q being points in 
c, C respectively. Then the quadrilateral qQPp is inscribed in a 
circle (that in which its plane cuts the sphere, J) and the exterior 
angle at p is equal to the interior angle at Q. If OL, OMhe the 
lines in which the plane POQ cuts the cyclic planes (planes through 
parallel to the two series of circular sections) they are obviously 
parallel to pq, QP, respectively ; and therefore 

LLOp = LOpq = loqp = iMoq. 

Let any third side, 
OrE, of the cone be 
drawn, and let the 
plane OPR cut the 
cyclic planes in 01, Om 
respectively. Then, 

L10L= L qpr, 

and these angles are independent of the position of the points p and 
P, if Q and R be fixed points. 

In a section of thp above 
diagram by a sphere whose 
centre is 0, IL, Mm are the 
great circles which repre- 
sent the cyclic planes, PQ,R 
is the spherical conic which 
represents the cone. The 
point P represents the line OpP, and so with the others. The 
propositions above may now be stated thus 
Arc PL = arc MQ ; 
and, if Q and R be fixed, Mm and IL are constant arcs whatever be 
the position of P. 




60.] The application to § 58 is now obvious. In the figure of 
that article we have 

FE=ES, EI) = CA, Sg = CB, LM = FI). 
Hence L, C, G, D are points of a spherical conic whose cyclic 
planes are those of AJB, FE. Hence also KG passes through L, 
and with LM intercepts on AB an arc equal to AB. That is, it 
passes through N, or KG and LN are arcs of the same great circle : 
and they are equal, for G and L are points in the spherical conic. 

Also, the associative principle holds for any number of quaternion 
factors. For, obviously, 

qr.st = qrs.t = &c., &c., 
since we may consider qr as a single quaternion, and the above 
proof applies directly. 

61.] That quaternion addition, and therefore also subtraction, 
is commutative, it is easy to shew. 

For if the planes of two quaternions, 
q and r, intersect in the line OA, we 
may take any vector OA in that line, 
and at once find two others, OB and 
OC, such that 

0B= qOA, 
and OC=r OA. 

And {q + r)0A=0B+0C=0C+0B={r+q)6A, 
since vector addition is commutative (§ 27). 

Here it is obvious that {q + r)OA, being the diagonal of the 
parallelogram on OB, OC, divides the angle between OB and OC 
in a ratio depending solely on the ratio of the lengths of these 
lines, i. e. on the ratio of the tensors of q and r. This will be useful 
to us in the proof of the distributive law, to which we proceed. 

62.] Quaternion multi- 
JD plication, and therefore di- 

vision, is distributive. One 
simple proof of this depends 
on the possibility, shortly to 
be proved, of representing 
an^ quaternion as a linear 
function of three given rect- 
angular unit-vectors. And 
when the proposition is thus 
established, the associative principle may readily be deduced froin it. 
But we may employ for its proof the properties of Spherical 


Conies already employed in demAsttating the truth of the asso- 
ciative principle. For continuity we give an outline of the proof 
by this process. 

Let BA, GA represent the versors of q and r, and be the great 
circle whose plane is that of js. 

Then, if we take as operand the vector OA, it is obvious that 
U{q + r) will be represented by some such arc as BA where B, B, C 
are in one great circle ; for {q + r) OA is in the same plane as q OA 
and rOA, and the relative magnitudes of the arcs BB and BC 
depend solely on the tensors of q and r. Produce BA, BA, CA to 
meet be in b, d, e respectively^ and make 

^ M = BA, m= BA, Gc= CA. 
Also make b^ = dh = cy =^. Then E, F, G, A lie on a spherical 
conic of which BG and be are the cyclic arcs. And, because 
bfi = dh = cy, pE, hF, y G, when produced, meet in a point R 
which is also on the spherical conic (§ 59*). Let these arcs meet 
BG in J, B, K respectively. Then we have 

JHz= E^ = pUq, 
LH=M =pU{q + r), 
KE= Gy =p Ur. 
Also fj= BB, 

and EL = CB. 

And, on comparing the portions of the figure bounded respectively 
by HKJ and by AGB we see that (when considered with reference 
to their effects as factors multiplying OH and OA respectively) 

J) U(^qjf.r) bears the same relation to jo Uq and jo Ur 
that lf{q + r)\)G2iXsto Uq&xA Ur. 
But T{q + r)U{q + r) = q + r = TqUq + TrUr. 

Hence T^ + r).jpU{q + r) = Tq.p Uq + Tr.p Ur ; 

or, since the tensors are mere numbers and commutative with all 
other factors, p{q + r) = pq +pr. 

In a similar manner it may be proved that 
{q + r)p = qp + rp. 
And then it follows at once that 

(p + q) (r+s) =pr +ps + qr-j-qs. 
63.1 By similar processes to those of § 53 we see that versors, 
and therefore also quaternions, are subject to the index-law 

q'^.q" = j'"+", 
at least so long as m and n are positive integers. 



The extension of this property to negative and fractional ex- 
ponents must be deferred until we have defined a negative or 
fractional power of a quaternion. 

64.] We now proceed to the special case of guadrantal versors, 
from whose properties it is easy to deduce all the foregoing results 
of this chapter. These properties were indeed those whose in- 
vention by Hamilton in 1843 led almost intuitively to the esta- 
blishment of the Quaternion Calculus. We shall content ourselves 
at present with an assumption, which will be shewn to lead to 
consistent results-; but at the end of the chapter we shall shew 
that no other assumption is possible, following for this purpose a 
very curious quasi-metaphysical speculation of Hamilton. 

65.] Suppose we have a system of three mutually perpendicular 
unit- vectors, drawn from one point, which we may call for short- 
ness I, J, K. Suppose also that these are so situated that a positive 
(i. e. left-handed) rotation through a right angle about / as an axis 
brings J to coincide with K. Then it is obvious that positive 
quadrantal rotation about / will make K coincide with /; and, 
about K, will make I coincide with /. 

For definiteness we may suppose J to be drawn eastwards, J north- 
wards, and K upwards. Then it is obvious that a positive (left- 
handed) rotation about the eastward line (7) brings the northward 
line (i7) into a vertically upward position {K); and so of the others. 

66.] Now the operator which turns J into Z" is a quadrantal 
versor (§ 53) ; and, as its axis is the vector I, we may call it i. 

Thus T'^^' °^ K=iJ- (1) 

Similarly we may put -= =j, or I=.jK, (2) 

and -Y = k, or J = hi. (3) 

[It may here be noticed, merely to shew the symmetry of the 
system we are explaining, that if the three mutually perpendicular 
vectors /, /, Xbe made to revolve about a line equally inclined to 
all, so that / is brought to coincide with J, J will then coincide 
with K, and X with I: and the above equations will still hold good, 
only (1) will become (2), (2) will become (3), and (3) will become 


67.] By the results of § 50 we see that 
-/_ K 
K ~ J' 




1. e. a southward unit- vector bears the same ratio to an upward 
unit-vector that the latter does to a northward one; and therefore 
we have 

~" -J=iK. 

— K 

Similarly -^^ = 





t, or 


= ff, or 







68.] By (4) and (1) we have. 

-J =iK= i{iJ) = i^J. 

Hence p = _ 1 

And, in the same way, (5) and (2) give 

-^'=-1' (8) 

and (6) and (3) F ^ — 1 (9^ 

Thus, as the directions of /, J, K are perfectly arbitrary, we see 
that Ue square of every quad/rantal versor is negative unity. 

Though the following proof is in principle exactly the same as 
the foregoing, it may perhaps be of use to the student, in shewing 
him precisely the nature as well as the simplicity of the step we 
have taken. 

Let ABA' be a semicircle, whose centre 
is 0, and let OB be perpendicular to AOA'. 

Then ^=:^ , = q suppose, is a quadrantal 



versor, and is evidently equal to -:=r; 
§§ 50, 53. _^ _ ^^ 

OA' OB 61' 

A' -ot 


r = -=- 



69.] Having thus found that the squares of i, J, h are each equal 
to negative unity ; it only remains that we find the values of their 
products two and two. For, as we shall see, the result is such as 
to shew that the value of any other combination whatever of i, j, Jc 
(as factors of a product) may be deduced from the values of these 
squares and products. 
Now it is obvious that 


-I~ K 
D a 

_I__ . 


(i. e. the versor which turns a westward unit- vector into an upward 
one will turn the upward into an eastward unit) ; 

or K = J{-I)=-jI* (10) 

Now let us operate on the two equal vectors in (10) by the same 
versor, i, and we have 

iK = i {—jl) = —ijl. 
But by (4) and (3) 

iK = -J =-kI. 
Comparing these equations, we have 
or, by § 54 (end), ij = k,^ 

and symmetry gives jh = i, \ (11) 

hi = j. J 
The meaning of these important equations is very simple ; and is, 
in fact, obvious from our construction in § 54 for the multiplication 
of versors; as we see by the annexed figure, where we must re- 
member that i, j, ^^re quadrantal versor s whose planes are at right 

angles, so that the figure represents a 
hemisphere divided into quadrantal tri- 

Thus, to shew that ij = k, we have, 
being the centre of the sphere, N, E, 
S, W the north, east, south, and west, 
and ^the zenith (as in § 65) ; 

j6W= 6z, 
whence ijOW=^iOZ= OS - kOW. 
70.] But, by the same figure, 

i_ON=OZ, _ _ _ 

whence jiON = jOZ= OE = -OW=-kON. 

71. J From this it appears that 

ji=-k, \ 

and similarly kj =- — i, > (12) 

ik = -J, ) 
and thus, by comparing (11), 

(/ = -i* = ^> ) 
jk=~kj=iA ((11), (12)). 

ki = —ik = J. ) 

* The negative sign, being a mere numerical &ctor, is evidently commutative with 
j ; indeed we may, if necessary, easily assure ourselves of the fact that to turn the 
negative (or reverse) of a vector through a right (or indeed any) angle, is the same 
thmg ae to turn the vector through that angle and then reverse it. 


These equations, along with 

i2=/=F=-l ((7), (8), (9)), 
contain essentially the whole of Quaternions. But it is easy to see 
that, for the first group, we may substitute the single equation 

V^=-l, (13) 

since from it, by the help of the values of the squares of i, J, h, all 
the other expressions may be deduced. We may consider it proved 
in this way, or deduce it afresh from the figare above, thus 
hON= 6W, 
jkON= j6W= 6Z, 
ijhON= ijOW=^i6Z= 68 = -0N. 
72.] One most important step remains to be made, to wit the 
assumption -referred to in § 64. We have treated i,j, k simply as 
quadrantal versors ; and /, J, K as unit-vectors at right angles to 
each other, and coinciding with the axes of rotation of these versors. 
But if we collate and compare the equations just proved We have 

= .k, (11) 

.iJ= K, (1) 

\Ji=-k, (12) 

Ijl = -K, (10) 

with the other similar groups symmetrically derived from them. 
Now the meanings we have assigned to i, _;', k are quite inde- 
pendent of, and not inconsistent with, those assigned to I, J, Ki 
And it is superfluous to use two sets of characters when one will 
suffice. Hence it appears that «', /, k may be substituted for J, /, K; 
in other words, a unit-vector when employed as a factor may be con^ 
sidered as a quadrantal versor whose plane is perpendicular to the 
vector. This is one of the main elements of the singular simplicity 
of the quaternion calculus. 

73.] Thus the product, and therefore the quotient, of two perpen- 
dicular vectors is a third vector perpendicular to hoth. 

Hence the reciprocal (§ 51) of a vector is a vector which has the 
opposite direction to that of the vector, and its length is the re- 
ciprocal of the length of the vector. 

The conjugate (§ 52) of a vector is simply the vector reversed. 

Hence, by § 52, if a be a vector 

{Taf = aKa = a (-a) = -a". 

74.J We may now see that every versor may be represented by 
a power of a unit-vector. 


For, if a be any vector perpendicular to i (which is an^ definite 

»a, = /3, is a vector equal in length to a, but perpendicular 
to both i and a ; 

i^a = — a, 
i^a =—ia = — /3, 
i*a = — i/3 = —i^ a = a. 
Thus, by successive applications of i, a is turned round i as an axis 
through successive right angles. Hence it is natural to define i*" as 
a versor which turns any vector perpendicular to i through m right 
angles in the positive direction of rotation about i as an axis. Here m 
may have any real value whatever, whole or fractional, for it is 
easily seen that analogy leads us to interpret a negative value of m 
as corresponding to rotation in the negative direction. 

75.] From this again it follows that any quaternion may be 
expressed as a power of a vector. For the tensor and versor elements 
of the vector may be so chosen that, when raised to the same power, 
the one may be the tensor and the other the versor of the given 
quaternion. The vector must be, of course, perpendicular to the 
plane of the quaternion. 

76.] And we now see, as an immediate result of the last two 
sections, that the index-law holds with regard to powers of a 
quaternion (§ 63). 

77.] So far as we have yet considered it, a quaternion has been 
regarded as the product of a tensor and a versor : we are now to 
consider it as a sum. The easiest method of so analysing it seems 
to be the following. 

Let ^=- represent any quaternion. Draw 

BC perpendicular to OA, produced if neces- 

Then, §19, OB = OC+CB. 
But, § 22, OC = xOA, 

where a; is a number, whose sign is the same 
as that of the cosine of Z AOB. 
Also, § 73, since CB is perpendicular to OA, 
CB = yOA, 
where y is a vector perpendicular to OA and CB, i.e. to the plane 
of the quaternion. 

TT OB (vOl + yOA 

Hence -^^^ = =i — =a! + v. 



Thus a quaternion, in general, mSy be decomposed into the sum of 
two parts, one numerical^ the other a vector. Hamilton calls them 
the SCALAE, and the vector, and denotes them respectively by the 
letters S and T prefixed to the expression for the quaternion. 
78.] Hence q = Sq+ Vq, and if in the above example 



then OB = 0C+ CB = Sq.OA-\- Fq.Ol*. 

The equation above gives 

GB = rq.OA. 
79.] If, in the figure of last section, we produce BG to B, so as 
to double its length, and join OB, we have, by § 52, 

^=Kq = SKq^YKq; 

.-. 6B=0C + CB = 8Kq.62+rKq.0J. 
Hence OC = SKq.OA, 

and CB = rKq.OA. 

Comparing this value of OC with that in last section, we find 

8Kq = 8q, (1) 

or the scalar of the conjugate of a quaternion is equal to the scalar of 
the quaternion. 
Again, CB — — CB by the figure, and the substitution of their 

values gives VKq^-Vq, '. (2) 

or the vector of the conjugate of a quaternion is the vector of the 
quaternion reversed. 

We may remark that the results of this section are simple con- 
sequences of the fact that the symbols S, V, K are commutative f. ^ 
Thus SKq = K8q = Sq, 

since the conjugate of a number is the number itself; and 
VKq = KVq = -rq{\ 73). 

* The points are inserted to shew that S and Y apply only to q, and not to qOA . 

+ It is curious to compare the properties of these quaternion symbols with those of 
the Elective Symbols of Logic, as given in Boole's wonderful treatise on the LoAm of 
Thought; and to think that the same grand science of mathematical analysis, by 
processes remarkably similar to each other, reveals to ns truths in the science of 
position far beyond the powers of the geometer, and truths of deductive reasoning to 
which unaided thoug'ht could never have led the logician. 


Again, it is obvious that 

:^Sq = S2q, l,Fq= Flq, 
and thence SKq = Kl,q. 

80.] Since any vector whatever may be represented by 
xi + yj-i-zl! 
where x, y, z are numbers (or Scalars), and i, j, h may be any three 
non-coplanar vectors, §§ 23, 25 — though they are usually under- 
stood as representing a rectangular system of unit-vectors — and 
since any scalar may be denoted by w; we may write, for any 
quaternion q, the expression 

q = w-\-m-\-yj->rzh (§ 78). 

Here we have the essential dependence on four distinct numbers, 
from which the quaternion derives its name, exhibited in the most 
simple form. 

And now we see at once that an equation such as 

where §^= v/+x'i-\-i^j-\-/h, 

involves, of course, the ybwr equations 

vf=w, af= X, y'=y, i^—z. 

81.] "We proceed to indicate another mode of proof of the dis- 
tributive law of multiplication. 

We have already defined, or assumed (§61), that 

- + - = > 

a a a 

or ^a-i+ya-i = (^ + y)a-i, 

and have thus been able to understand what is meant by adding 
two quaternions. 

But, writing o for a~^, we see that this involves the equality 
[fi + y)a = /3a + ya; 
from which, by taking the conjugates of both sides, we derive 

And a combination of these results (putting /3 + y for a' in the 
latter, for instance) gives 

(^ + y)(^+/) = (/3+y)^+G3 + y)/ 

= i3/3'-|- yi3'+ /3y'+ yy by the former. 
Hence the distributive principle is true in the multiplication of vectors. 
It only remains to shew that it is true as to the scalar and 


vector parts of a quaternion, and then we shall easily attain the 
general proof. 

Now, if a be any scalar, a any vector, and q^ any quaternion, 
(a + aj 3' = Ǥ' + aq. 
For, if ;3 be the vector in which the plane of §' is intersected by 
a plane perpendicular to a, we can find other two vectors, y and 8, 
in these planes such that 

And,. of course, a may be written ■—; so that 

^ ^^ y3 8 8 

-'^6 +6-'^8 + ^ 6 
= aq-\-aq^. 
And the conjugate may be written 

/(a' + o') = ?V + /a' (§55). 
Hence, generally, 

(a + a)(3 + ;8) =ffli + a/3 + Ja+a;8; 
or, breaking up « and 5 each into the sum of two scalars, and a, /3 
each into the sum of two vectors, 

(«i + aa + oi + Og) ( *! + «2 + ^1 + /Sg) 

= K + «2) (*i + ^2) + («i + «.) (/3i + /Sz) + {\ + ^2) («i + "2) 

+ («l + «2)(^l + /32) 

(by what precedes, all the factors on the right are distributive, so 
that we may easily put it in the form) 

= («i + «i) (^1 + /5i) + K + «i) (*2 + /32) + («2 + a^) («i + /3i) 

+ («2+a2)(52 + /32)- 

Putting ai + ai=j9, «2 + a2 = $', ^i + ZSj = /, 5a + /32=», 
we have ( J" + ?) 0" + *) =/'>" + i'»+$'»' + S'*. 

83.] For variety, we shall now for a time forsake the geometrical 
mode of proof we have hitherto adopted, and deduce some of our 
next steps from the analytical expression for a quaternion given 
in § 80, and the properties of a rectangular system of unit-vectors 
as in § 71. 

We will commence by proving the result of § 77 anew. 

83.] Let a = xi + yj-\-zh. 


Then, because by § 7 1 every product or quotient of i, J, h is reducible 
to one of them or to a number, we aye entitled to assume 

^ = ^ = a) + ^J + 77y+C^, 

where co, f, 77, f are numbers. This is the proposition of § 80. 

84.] But it may be interesting to find to, £, tj, f in terms of x, y, z, 
af,^, z. 

We have ^ = qa, 

or x'i-\-y'j-\-^k = (a, +^»4- v'+ C^) {xi + yj+zk) 

as we easily see by the expressions for the powers and products of 
hji '^j given in § 71. But the student must pay particular attention 
to the order of the factors, else he is certain to make mistakes. 
This (§ 80) resolves itself into the four equations 
0= ^SB + riy + Cm, 

x'=(x,x +i?«— fy, 

^=wy—iz +Ca?, 

/= a)Z + $y—r]X. 
The three last equations give 

xx'+yy' + zz'= a) {x^ + y^ + z^), 
which determines m. 
Also we have, from the same three, by the help of the first, 
ix' + riy'+C/=0; 
which, combined with the first, gives 

^ = ■n ^ C . 

y/—zy' zuZ—x/ x^—yx^' 
and the common value of these three fractions is then easily seen 
to be 1 

x'^+y^+ z^ 

It is easy enough to interpret these expressions by means of 
ordinary coordinate geometry : but a much simpler process will 
be furnished by quaternions themselves iti the next chapter, and, in 
giving it, we shall refer back to this section. 

85.] The associative law of multiplication is now to be proved 
by means of the distributive (§ 81). We leave the proof to the 
student. He has merely to multiply together the factors 

w + xi + yj + zk, w'+x'i + yj+Zk, and w" + x"i + y"j+/'k, 
as follows : — 

First, multiply the third factor by the second, and then multiply 
the product by the first ; next, multiply the second factor by the 


first and employ the product to multiply the third : always re- 
membering that the multiplier in any product is placed lefore the 
multiplicand. He will find the scalar parts and the coefficients of 
i,j, Te, in these products, respectively equal, each to each. 

86.] With the same expressions for a, )3, as in section 83, we have 
a^ = {xi + yj + zh) {x'i + y'j + ^h) 

= - [xx' +yy' + zz') + ly/ -zy')i + {zx'-x^)j-\- {xy" -yx')k. 
But we have also 

^a= — {xx'+ yy' +zz')- {y/ — z/) i - {zaf - xz')j — (a/ - yx) k 
The only difference is in the sign of the vector parts. 

Hence Sa^ = Spa, (1) 

ral3=-r^a, (2) 

also afi + ^a = 2Sa^, (3) 

al3-^a = 2rap, (4) 

and, finally, by § 79, a^ = K^a (5) 

87.] If a = /3 we have of course (§ 25) 

x = x', y=y', z = z', 
and the formulae of last section become 

a^ — ^a = a' = —(x' + y^ + z') ; 
vsrhich was anticipated in § 73, where we proved the formula 

and alsOj to a certain extent, in § 25. 

88.] Now let q and / be any quaternions, then 
S.qr = S.{Sq+ Vq) {Sr+ Fr), 

= S.{SqSr+Sr.Vq + Sq.Fr+ FqFr), 
= SqSr+SFqFr, 
since the two middle terms are vectors. 
Similarly, S.rq = SrSq + SFr Fq. 

Hence, since by (1) of § 86 we have 

SFqFr = SFrFq, 
we see that S.qr = S.rq, (1) 

a formula of considerable importance. 

It may easily be extended to any number of quaternions, because, 
r being arbitrary, we may put for it rs. Thus we have 
S.qrs = S.rsq, 
= S.sqr 
by a second application of the process. In words, we have the 
theorem — tAe scalar of ike product of any number of given quaternions 
depends only upon the cyclical order in which they are arranged. 


89.] An important case is that of three factor.s, each a vector. 
The formula then becomes 

S.a^y = S.^ya = S.ya^. 
But S. aPy = Sa{Spy + V^y) 

= SaVfiy, since aSfty is a vector, 
= -Saryp, by (2) of §86, 
= -Sa{SyP+ryP) 
= -S.ayfi. 
Hence tke scalar of the product of three vectors changes sign when the 
cyclical order is altered. 

Other curious propositions connected with this will be given 
later, as we wish to devote this chapter to the production of the 
fundamental formulae in as compact a form as possible. 
90.] By (4) of §86, 

Hence 2FaFfiy = Fa {fiy — y^) 

(by multiplying both by a, and taking the vector parts of each side) 

= F{apy+pay—pay—ayfi) 
(by introducing the null term ^ay—^ay). 
That is 

2rar^y=r.{aP + l3a)y-F(fiSay + l3Fay + Say.p+Fay.p) 
= F{2Sa^)y-2F^Say 
(if we notice that F.Fay.^ =~ F^Fay, by (2) of § 86). 

Hence FaF^y = ySa^—fiSya, (1) 

a formula of constant occurrence. 

Adding aS^y to both sides we get another most valuable formula 

F.afiy = aSfiy-^Sya + ySa^; (2) 

and the form of this shews that we may interchange y and o 
without altering the right-hand member. This gives 

F.a^y = F.y^a, 
a formula which may be greatly extended. 

91.] We have also 
FFa^Fyh = - FFyb Fafi by (2) of § 86 : 

= bSyFap-ySbFafi = bS.afiy-yS.a^b, 
= - pSa, Fyh + axS/3 Fyh = -pS. ayh + aS. j8y8, 
all of these being arrived at by the help of § 90 (1) and of § 89 ; 
and by treating alternately Fa/3 and Fyb as simple vectors. 
Equating two of these values, we have 

bS.al3y = aS.I3yb + l38.yab + yS.al3t, (3) 


a very useful formula, expressing any vector whatever in terms 
of three given vectors. 

93.] That such an expression is possible we knew already by 
§ 23. For variety we may seek another expression of a similar 
character^ by a process which differs entirely from that employed 
in last section. 

a, /3, y being any three vectors, we may derive from them three 
others Faj3, V^y, Vya ; and, as these will not generally be coplanar, 
any other vector 6 may be expressed as the sum of the three, each 
multiplied by some scalar (§ 23). It is required to find this ex- 
pression for 5. 

Let h=go FajS + i/Vfiy + z Vya. 

Then 8yb = xS.yajS = xS.a^y, 

the terms in ^ and z going out, because 

Syri3y = S.yPy = S/Sy^ = y^SjS = 0, 
for y^ is (§ 73) a number. 

Similarly /S/38 = zS.^ya = zS.aj3y, 
and Sab = yS.a^y. 
Thus bS.apy = Fa^Syb + FjSySab + VyaS^b (4) 

93.] We conclude the chapter by shewing (as promised in § 64) 
that the assumption that the product of two parallel vectors is 
a number, and the product of two perpendicular vectors a third 
vector perpendicular to both, is not only useful and convenient, 
but absolutely inevitable, if our system is to deal indifferently with 
all directions in space. We abridge Hamilton's reasoning. 

Suppose that there is no direction in space pre-eminent, and 
that the product of two vectors is something which has quantity, 
so as to vary in amount if the factors are changed, and to have 
its sign changed if that of one of them is reversed ; if the vectors 
be parallel, their product cannot be, in whole or in part, a vector 
inclined to them, for there is nothing to determine the direction 
in which it must lie. It cannot be a vector parallel to them ; for 
by changing the sign of both factors the product is unchanged, 
whereas, as the whole system has been reversed, the product vector 
ought to have been reversed. Hence it must be a number. Again, 
the product of two perpendicular vectors cannot be' wholly or partly 
a number, because on inverting one of them the sign of that 
number ought to change; but inverting one of them is simply 
equivalent to a rotation through two right angles about the other, 
and (from the symmetry of space) ought to leave the number 


unchanged. Hence the product of two perpendicular vectors must 
be a vector, and a simple extension of the same reasoning shews 
that it must be perpendicular to each of the factors. It is easy 
to carry this farther, but enough has been said to shew the character 
of the reasoning. 


1 . It is obvious from the properties of polar triangles that any 
mode of representing versors by the sides of a triangle must have 
an equivalent statement in which they are represented by angles in 
the polar triangle. 

Shew directly that the product of two versors represented by 
two angles of a spherical triangle is a third versor represented 
by the supplement of the remaining angle of the triangle; and 
determine the rule which connects the directions in which these 
angles are to be measured. 

2. Hence derive another proof that we have not generally 

m = IP- 

3. Hence shew that the proof of the associative principle, § 57, 
may be made to depend upon the fact that if from any point of 
the sphere tangent arcs be drawn to a spherical conic, and also arcs 
to the foci, the inclination of either tangent arc to one of the focal 
arcs is equal to that of the other tangent arc to the other focal arc. 

4. Prove the formulae 

2S.apy = a^y—y^a, 
2r.a/3y= a^y + y^a. 

5. Shew that, whatever odd number of vectors be represented by 
a, 13, y, &c., we have always 

F.aPybe = V.eby^a, 

KajaybeCn = r.7jfe8y/3a, &c. 

6. Shew that 

S.FajSFfiY^ya = -{S-ajSyf, 

r. ral3 r^y Vya = TajS {y^Sa^ - SfiySya) + , 

and F. ( Fa^ F. Ffiy Fya) = (fiSay- aSfiy) S.a^y. 

7. If a, /3, y be any vectors at right angles to each other, shew that 

(a3 + ^3 + y 3) ;S.a^y = a* Fj3y + /3* Fya + y* Fafi. 


8. If a, j3, y be non-eoplanar veifcors, find the relations among 
the six scalars, x, y, z and f, t\, f, which are implied in the equation 

xa->ry^-^zy = iFfSy + r\ Fya + QFa^. 

9. If a, j3, y he any three non-eoplanar vectors, express any 
fourth vector, 6, as a linear function of each of the following sets of 
three derived vectors, 

r.yap, V.afiy, V.^ya, 
and V.ra^r^yVya, FV^yVyaVa^, F.FyaFapFfiy. 

10. Eliminate p from the equations 

Sap = a, Sj3p = b, Syp = c, Sbp = d, 
where a, /3, y, 6 are vectors, and a, b, c, d scalars. 

11. In any quadrilateral, plane or gauche, the sum of the squares 
of the- diagonals is double the sum of the squares of the lines joining 
the middle points of opposite sides. 



94.] Among the most useful characteristics of the Calculus of 
Quaternions, the ease of interpreting its formulae geometrically, 
and the extraordinary variety of transformations of which the 
simplest expressions are susceptible, deserve a prominent place. 
We devote this Chapter to some of the more simple of these, to- 
gether with a few of somewhat more complex character but of 
constant occurrence in geometrical and physical investigations. 
Others will appear in every succeeding Chapter. It is here, 
perhaps, that the student is likely to feel most strongly the peculiar 
difficulties of the new Calculus. But on that very account he 
should endeavour to master them, for the variety of forms which 
any one formula may assume, though puzzling to the beginner, is 
of the most extraordinary advantage to the advanced student, not 
alone as aiding him in the solution of complex questions, but as 
affording an invaluable mental discipline. 

95.] If we refer agiain to the figure of § 77 we see that 
0C= OB cos AOB, 
CJB = OB sin JOB. 

Hence, if OJ = a, OB = p, and /.AOB = 6, we have 

OB = Tl3, OA = Ta, 

OC = Tfi cos d, CB = Tfi sin 0. 

o/3 OC Tl3 
Hence S- = -^-r = -y==— cos5. 

a OA la 

Similarly rr^ = ^ = ^sin.. 


Hence, if e be a unit- vector perpenoicular to o and /3, or 

UOA o. 

we have F- = -^ sin 0.{. 

a Ta 

96.] In the same way we may shew that 

TVa^ = Ta Tfi sin 6, 
and Fa^ = Ta Tfi sin O.r, 

where 77= Urap = UF^- 


Thus tAe scalar of the product of two vectors is the continued product 
of their tensors and of the cosine of the sitpplement of the contained 

The tensor of the vector of the product of two vectors is the con- 
tinued product of their tensors and the sine of the contained angle ; 
and the versor of the same is a unit-vector perpendicular to both, and 
such that the rotation about it from the first vector (i. e. the multiplier') 
to the second is left-handed or positive. 

Hence TFa^ is doMe the area of the triangle two of whose sides 
are a, /3. 

(a.) In any triangle A£C we have 
AC = A£ + W. 
Hence IC^ = SAC AC = S.AC{AB + SC). 

With the usual notation for a plane triangle the interpretation 
of this formxila is 

—b^ = —be cos A— ab cos, C, 
or b= a cos C+c cos A 

(b.) Again we have, obviously, 

rABAC= FAS{A£ + BC) 
or cb sin A = ca sin B, 

sin A sin B sin C 

whence = — j— = • 

a c 

These are truths, but not truisms, as we might have been led 

to fancy from the excessive simplicity of the process employed. 



98.] From § 96 it follows that, if a and /3 be both actual (i. e. 
real and non-evanescent) vectors, the equation 

Sa^ = 
shews that cos 6 = 0, or that a is perpendicular to /3. And, in fact, 
we know already that the product of two perpendicular vectors is a 

Again, if ^„^ = 0, 

we must have sin ^ = 0, or a is parallel to /3. We know already 
that the product of two parallel vectors is a scalar. 
Hence we see that 

Sa^ = 

is equivalent to o = Fy/3, 

where y is an undetermined vector ; and that 

is equivalent to a = a;/3, 

where aj is an undetermined scalar. 

99.] If we write, as in § 83, 

o — ix +ji/ + kz, 

/3 = ix'+j/ + M, 

we have, at once, by § 86, 

Sa^ = —xaf—yy'—z/ 

, ^x af , y 1/ z z' \ 

= -r/( - + I-£^ + — ) 

\ r r r r r r ' 

where r = -s/as^+j^^ +«^ /= \/«'^+y^+/^. 

Also r^ = ^|i:!:^i+ff:^-y+at.er,j. 

These express in Cartesian coordinates the propositions we have 
just proved. In commencing the subject it may perhaps assist 
the student to see these more familiar forms for the quaternion 
expressions ; and he will doubtless be induced by their appearance 
to prosecute the subject, since he cannot fail even at this stage to 
see how much more simple the quaternion expressions are than 
those to which he has been accustomed. 

100.] The expression S.a&y 

may be written S ( Fa/3) y, 

because the quaternion a.^y may be broken up into 

of which the first term is a vector. 


But, by §96, 

S ( ra/3) y = TaTl3 sin 9 Sriy. 

Here Tr) = 1, let (|) be the angle between ?j and y, then finally 

S.a^y =-TaT^Ty sin 5 cos (|). 
But as ?j is perpendicular to a and /3, Ty cos ^ is the length of the 
perpendicular from the extremity of y upon the plane of a, /3. And 
as the product of the other three factors is (§ 96) the area of the 
parallelogram two of whose sides are a, ^, we see that the mag- 
nitude otS.apy, independent of its sign, is i^e volume of the parallel- 
epiped of which three coordinate edges are a, fi, y; or six times the 
volume of the pyramid which has a, ^, y for edges. 
101.] Hence the equation 

S.apy = 0, 
if we suppose a, /S, y to be actual vectors, shews either that 

sin e = 0, 
or cos(^ = 0, 

i. e. two of the three vectors are parallel, or all three are pArallel to 
one plane. 

This is consistent with previous results, for if y = ^j3 we have 
S.aPy=:pS.afi^ = Q; 
and, if y be coplanar with a, fi, we have y =pa + qP, and 
S.al3y = S.al3{pa + ql3) = 0. 
102.] This property of the expression S.a^y prepares us to find 
that it is a determinant. And, in fact, if we take a, ;3 as in § 83, 
and in addition ^ ^ ^^" +_^y ^ ^/'^ 

we have at once 

S.apy = —x" [yi^-zy')-f {zx'—xz) ^z" {x/ —yx'), 
=.— X y z 
x' y' / 
of' f z" 

The deterrhinant changes sign if we make any two rows change 
places. This is the proposition we met with before (§ 89) in the 
form s^afiy = ^S.jSay = S.^ya, &e. 

If we take three new vectors 

ai = ix+j'a^+^a/', 

yi = iz+J/+M', 
we thus see that they are coplanar if o, ;3, y are so. That is, if 

iS.al3y = 0, 
then (S.Oj/Sjyi = 0. 

E -2 

52 QUATERNIONS. [103. 

103.] We have, by § 52, 

{Tqf = qKq = {Sq+ Fq) (Sq- fq) (§ 79), 
= lSqf-{rqf by algebra, 
= {SqY+{Trqf (§73). 
liq = aj8, we have Kq = fia, and the formula becomes 

a/3.;8a = a''^^ = {Sa^f-{Va^f. 
In Cartesian coordinates this is 


More generally we have 

(r(gr))2 = qrK{qr) 

= qrKrKq (§ 55) = {Tqf {Trf (§ 52). 
If we write q =.w ■\-a = w +ix +jy + kz, 

r = w' + l3 =. w'+iaf+Jy'+k/; 
this becomes 

= {wio'—xx'—^^'—z/f + {loixf + «/«; +^/—z/)'^ 
+ {w/ + 'u/y+zx'—x/y + {10/ +w'z+x/—ya/)^, 
a formula of algebra due to Euler. 

104.] We have, of course, by multiplication, 

(a+/3)2 = a^ + aj3 + ^a + P^ = a' + 2Sa^ + fi'' (§86 (3)), 
Translating into the usual notation of plane trigonometry, this 
becomes c^ =za^-2ah cos C+ b% 

the common formula. 

Again, r(a+/3) (a-/3) = - rai3+ T/Sa = -2 FaiS (§ 86 (2)). 
Taking tensors of both sides we have the theorem, the jparallelogram 
whose sides are parallel and equal to the diagonals of a given paral- 
lelogram, has double its area (§ 96). 

Also iS(a + /3)(a-/3) = a^-^^ 

and vanishes only when a^ = /3^, or Ta—T^\ that is, the diagonals 
of a parallelogram are at right angles to one another, when, and only 
when, it is a rhombus. 

Later it will be shewn that this contains a proof that the angle in 
a semicircle is a right angle. 

105.] The expression p = a^a'^ 

obviously denotes a vector whose tensor is equal to that of /3. 

But we have S.^ap = 0, 

so that p is in the plane of o, ^. 

Also we have Sap = Sa^, 


so that /3 and p make equal angles with a, evidently on opposite 
sides of it. Thus if a be the perpendicular to a reflecting surface 
and /3 the path of an incident ray, p will be the path of the re- 
flected ray. 

Another mode of obtaining these results is to expand the above 
expression, thus, § 90 (2), 

p = 2a-^Sa^-^^ 

SO that in the figure of § 77 we see that if OA = a, and OJB = ^, we 
have OJ) = p = a^a~''^. 

Or, agaiuj we may get the result at once by transforming the 

equation to U- = U-- 

106.] For any three coplanar vectors the expression 
p = afiy 
is (§ 101) a vector. It is interesting to determine what this vector 
is. The reader will easily see that if a circle be described about 
the triangle, two of whose sides are (in order) a and /3, and if from 
the extremity of /3 a line parallel to y be drawn again cutting the 
circle, the vector joining the point of intersection with the origin 
of a is the direction of the vector afiy. For we may write it in the 

form a 

p = a^^fi-^y = -{T^fafi-^y = -{T^f -y, 

which shews that the versor (-A which turns j3 into a direction 

parallel to a, turns y into a direction parallel to p. And this ex- 
presses the long-known property of opposite angles of a quadri- 
lateral inscribed in a circle. 

Hence if a, ^, y be the sides of a triangle taken in order, the 
tangents to the circumscribing circle at the angles of the triangle 
are parallel respectively to 

a^y, Pya, and ya)3. 
Suppose two of these to be parallel, i. e. let 
a/3y = x^ya = as ay 13 (§ 90), 
since the expression is a vector. Hence 

Py = xyp, 
which requires either 

x=\, Fy^ = or y || /3, 
a case not contemplated in the problem ; 

or a; = -l, S^y = 0, 

54 QUATERNIONS, [107. 

i. e. the triangle is right-angled. And geometry shews us at once 
that this is correct. 

Again, if the triangle be isosceles, the tangent at the vertex is 
parallel to the base. Here we have 

wfi = a^y, 
or (X!{a + y) = a{a + y)y; 
whence x = y'' = a?, or Ty = Ta, as required. 

As an elegant extension of this proposition the reader may prove 
that the vector of the continued product a^yS of the vector-sides of 
a quadrilateral inscribed in a sphere is parallel to the radius drawn 
to the corner {a, 8). 

107.] To exemplify the variety of possible transformations even 
of simple expressions, we will take two cases which are of frequent 
occurrence in applications to geometry. 

Thus T{p-\-a) = T{s>-a), 

[which expresses that if 

02 = a, 0A'= —a, and OP = p, 
we have AP = A'F, 

and thus that P is any point equidistant from two fixed points,] 
may be written (p + a)^ = {p—af, 

or p'^ + iSap + a^ = p^ — ^Sap + a^ {^101), 
whence Sap = 0. 

This may be changed to 

ap+ pa = 0, 
or ap + Kap = 0, 

SU^ = 0, 

or finally, TFU^ = 1, 

all of which express properties of a plane. 
Again, Tp = Ta 

may be written T - = 1, 

^ a'' ^ a'' 

(p + aY-28a{p + a) = 0, 

p= {p + a)-'^a{p+a), 

S{p + a){p—a) = 0, or finally, 

T.{p + a){p-a) = 2TVap. 



All of these express properties of a sphere. They will be in- 
terpreted when we come to geometrical applications. 

108.] "We have seen in § 95 that a quaternion may be divided 
into its scalar and vector parts as follows : — 

a a a Ta 

where 9 is the angle between the directions of a and /3, and e= UF- 


is the unit- vector perpendicular to the plane of a and /3 so situated 

that positive (i. e. left-handed) rotation about it turns a towards /3. 

Similarly we have (§ 96) 

0/3 = Sa^ + Fa^ 

= TaT^{-cose + esin0), 
6 and e having the same signification as before. 

109.] Hence, considering the versor parts alone, we have 

U- = cos6 + t sin d. 

Similarly U^ = cos (j) + e sincj} ; 

(j) being the positive angle between the directions of y and /3, and e 
the same vector as before, if a, /3, y be coplanar. 
Also we have 

U- = cos {d + <t)) + e sin {6 + <(>). 
But we have always 

-•- = -, and therefore 
|3 a a 

pa a 

or cos (<^ + 5) + e sin ((/)-}- 5) = (cos ^ -f e sin ^) (cos 5 + e sin 0) 

= cos (\) cos 5— sin (^ sin 9 + e (sin (pcos6 + cos ^ sin 6), 
from which we have at once the fundamental formulae for the 
cosine and sine of the sum of two arcs, by equating separately the 
scalar and vector parts of these quaternions. 

And we see, as an immediate consequence of the expressions 

abovcj that 

cos me + esmme = (cos -f e sin BJ" 

if m be a positive whole number. For the left-hand side is a versor 
which turns through the angle m5 at once, while the right-hand 


side is a versor which effects the same object by m successive turn- 
ings each through an angle Q. See § 8. 

110.] To extend this proposition to fractional indices we have 

only to write - for Q, when we obtain the results as in ordinary 


From De Moivre's Theorem, thus proved, we may of course 
deduce the rest of Analytical Trigonometry. And as we have 
already deduced, as interpretations of self-evident quaternion trans- 
formations (§§97, 104), the fundamental formulae for the solution 
• of plane triangles, we will now pass to the consideration of spherical 
trigonometry, a subject specially adapted for treatment by qua- 
ternions ; but to which we cannot afford more than a very few 
sections. (More on this subject will be found in Chap. X, in con- 
nexion with the Kinematics of rotation.) The reader is referred to 
Hamilton's works for the treatment of this subject by quaternion 

111. J Let a, /3, y be unit-vectors drawn from the centre to the 
corners A, JB,C oi a triangle on the unit-sphere. Then it is evident 
that, with the usual notation, we have (§ 96), 

Sa^ = — cos c, Sfiy = —cos a, Sya = —cos &, 

Trap= sine, TF^y = sin«, TFya= sin 3. 
Also UVafi, UFj3y, UFya are evidently the vectors of the corners 
of the polar triangle. 

Hence S. UFa^ UF^y = cos £, &c., 

TF.UFa^UF^y = BinB, &c. 
Now (§ 90 (1)) we have 

SFapFpy = S.aF.^Fpy 

=:-Sal38fiy + ^^Say. 
Remembering that we have 

SFa^F^y = TFa^TF^yS.UFapUF^y, 
we see that the formula just written is equivalent to 
sin a sin c cos B — ■— cos a cos c + cos h, 
or cos h = cos a cos c + sin a sin o cos B. 
112.] Again, F.Fa^F^y = -fiSa^y, 
which gives 

TF. FapF^y = S.apy = S.aFfiy = S.^Fya = S.yFa^, 
or sin a sin csinB = sin a sin^„ = sin b sin p^ = sin c sinjO„ ; 
where ^„ is the arc drawn from A perpendicular to BC, &c. 


Hence sin jo„ = sin e sin £, 

sin a sin c . _ 

sm Ml = -. — 5 — sm />, 


sin^o = sin a sin S. 

113.] Combining the results of the last two sections, we have 

Va^.V^y = sin a sin c cos 5— ^ sin a sine sin 5 

= sina sine (cos^— /3 sin 5). 

Hence U. Va^ V^y — (cos 5—^3 sin B), 1 

and U. Fy^r^a = (cos ^+ i3 sin B). ) 

These are therefore versors which turn the system negatively or 
positively about 0£ through the angle £. 
As another instance, we have 

sin 5 

tan^ = 

cos 5 

_ Tr.Va^r^y 

~ S.Va^r^y 

_ r.ra^rfiy 
'^ s.ra^r^y 

Say + SafiSfiy 

The interpretation of each of these forms gives a different theorem 
in spherical trigonometry. 

Again, let us square the equal quantities 

F. ajSy and cuS^y— jSSay + ySa^, 

supposing a, jS, y to be any unit- vectors whatever. We have 

-{KajSyY = S^^y + S^ya + S^afi+2SfiySyaSafi. 

But the left-hand member may be written as 


1-S^.a^y = S^fiy + S^ya + S^afi + 2S^ySyaSa^, ■ 

or 1 — cos^fl! — cos^S — cos^c + 2 cos a cos i cos c 

= sin^a sin^jo„ = &c. 

^ sin^asin^3sin^C= &c., 

all of which are well-known formulae. 

Such results may be multiplied indefinitely by any one who has 
mastered the elements of quaternions. 




114.] A curious proposition, due to Hamilton, gives us a qua- 
ternion expression for the spherical excess in any triangle. The 
following proof, which is very nearly the same as one of his, though 
by no means the simplest that can be given, is chosen here because 
it incidentally gives a good deal of other information. We leave 
the quaternion proof as an exercise. 

Let the unit- vectors drawn from the centre of the sphere to 
A, B, C, respectively, be a, p, y. It is required to express, as an 
arc and as an angle on the sphere, the quaternion 

The figure represents an orthographic projection made on a plane 
perpendicular to y. Hence G is the centre of the circle BEe. Let 
the great circle through A, B meet BBe in E, e, and let BB be a 
quadrant. Thus 2?^ represents y (§ 72). Also make BF=AB=pa~\ 
Then, evidently, ^ ^ ^a-^y, 

which gives the arcual representation required. 

Let BF cut Be in G. Make Ga = EG, and join B, a, and a, F. 
Obviously, as B is the pole of Ee, Ba is a quadrant ; and since 
EG — Ca, Ga = EG, a quadrant also. Hence a is the pole oi BG, 
and therefore the quaternion may be represented by the angle BaF. 

Make C6 = Ga, and draw the arcs P«/3, Pba from P, the pole of 
AB. Comparing the triangles Eba and ea(3, we see that Ea = e/3. 
But, since P is the pole of AB, F^a is a right angle : and therefore 
as i''a is a quadrant, so is F^. Thus AB is the complement of Ba. 
or ySe, and therefore „o _ lAB. 


Join bA and produce it to c so tnat Ac = hA; join e, P, cutting 
AS in 0. Also join c, £, and £, a. 

Since Pis the pole of AS, the angles at o are right angles ; and 
therefore, by the equal triangles 6aA, go A, we have 

aA = Ao. 
But a^ = 2AB, 

whence oB = B^, 

and therefore the triangles coB and Bafi are equal, and c, ^, a lie 
on the same great circle. 

Produce cA and cB to meet in M (on the opposite side of the 
sphere). H and c are diametrically opposite, and therefore cP, 
produced, passes through H. 

Now Pa = Pb = PH, for they differ from quadrants by the equal 
arcs fl/3, ba, oc. Hence these arcs divide the triangle Eab into three 
isosceles triangles. 

But IPHb + IPHa = LaHb = Ibca. 

Also /.Pab = TT—Zcab — Z-PaH, 

LPba =. LPab = it- Lcba- LPbH. 
Adding, iLPab^lis— Leah — Lcba— Lbca 

= IT — (spherical excess oi abc). 
But, as LFaj3 and LBae are right angles, we have 

angle of /3a~V = ^^aJ) = L^ae — LPab 

= \ (spherical excess, of abc). 

[Numerous singular geometrical theorems, easily proved ab initio 
by quaternions, follow from this : e. g. The arc AB, which bisects 
two sides of a spherical triangle abc, intersects the base at the 
distance of a quadrant from its middle point. All spherical tri- 
angles, with a common side, and having their other sides bisected 
by the same great circle (i. e. having their vertices in a small circle 
parallel to this great circle) have_equal areas, &e., &c.J 

115.] Let 0« = a, Ob = /3', Oc = y', and we have 


^^V Vy'. 

= Ca.BA 
But FQ is the complement of BF. Hence the angle of the 

quaternion , a A ^ /S'v I / /v 

Kj') yz') \7) 

60 QUATERNIONS. [ll6. 

is half the spherical excess of the triangle whose angular points are at 
the extremities of the unit-vectors a', ^', y' . 

[In seeking a purely quaternion proof of the preceding proposi- 
tions, the student may commence by shewing that for any three 
unit- vectors we have a.,„ 

The angle of the first of these quaternions can be easily assigned ; 
and the equation shews how to find that of /Sa-^y. But a stUl 
simpler method of proof is easily derived from the composition of 

116.] A scalar equation in p, the vector of an undetermined 
point, is generally the equation of a surface; since we may sub- 
stitute for p the expression . _ ^^j 

where x is an unknown scalar, and a any assumed unit-vector. 
The result is an equation to determine x. Thus one or more points 
are found on the vector xa whose coordinates satisfy the equation j 
and the locus is a surface whose degree is determined by that of the 
equation which gpives the values of x. 

But a vector equation in p, as we have seen, generally leads to 
three scalar equations, from which the three rectangular or other 
components of the sought vector are to be derived. Such a vector 
equation, then, usually belongs to a definite number oi points in 
space. But in certain cases these may form a line, and even a 
surface, the vector equation losing as it were one or two of the 
three scalar equations to which it is usually equivalent. 

Thus while the equation ap — & 
gives at once p _ „-i^^ 

which is the vector of a definite point (since we have evidently 

/Sa/3 = 0) ; 
the closely allied equation y^^ _ a 

is easily seen to involve g^o _ q 

and to be satisfied by p — oT'^R+xa 

whatever be x. Hence the vector of any point whatever in the line 
drawn parallel to a from the extremity of a~^/3 satisfies the given 

117.] Again, Fap .Fp^ = {FafiY 

is equivalent to but two scalar equations. For it shews that Fap 


and F)3p are parallel, i. e. p lies in fhe same plane as a and (3, and 
can therefore be written (§ 24) 

p = asa+^A 
where x and _y are scalars as yet undetermined. 
We have now Fap = yVafi, 

which, by the given equation, lead to 

xy =■ \, or y = -, or finally 

p = xa+~j3i 

which (§ 40) is the equation of a hyperbola whose asymptotes are 

in the directions of a and ^8. 

118.] Again, the equation 

r.raprap = o, 

though apparently equivalent to three scalar equations, is really 
equivalent to one only. In fact we see by § 91 that it may be 
written -aS.a^p = 0, 

whence, if a be not zero, we have 

S.ajSp = 0, 
and thus (§101) the only condition is that p is coplanar with a, j3. 
Hence the equation represents the plane in which o and )3 lie. 

119.] Some very curious results are obtained when we extend 
these processes of interpretation to functions of a quaternion, 

q = w+p 
instead of functions of a mere vector p. 

A scalar equation containing such a qtiaternion, along with 
quaternion constants, gives, as in last section, the equation of a 
surface, if we assign a definite value to w. Hence for successive 
values of w, we have successive surfaces belonging to a system; 
and thus when w is indeterminate the equation represents not a 
surface, as before, but a volume, in the sense that the vector of any 
point within that volume satisfies the equation. 

Thus the equation {Tqf = a^, 

or w'^—p^ = a^, 

or ' {Tpf = a^-w^, 

represents, for any assigned value of w, not greater than a, a sphere 
whose radius is ^/a^ — w^. Hence the equation is satisfied by the 

62 QUATEENIONS. [l20. 

vector of any point whatever in the volume of a sphere of radius a, 
whose centre is origin. 

Again, by the same kind of investigation, 

where q = w + p, is easily seen to represent the volume of a sphere 
of radius* a: described about the extremity of ^ as centre. 

Also S{^)-= —a? is the equation of infinite space less the space 
contained in a sphere of radius a about the origin. 

Similar consequences as to the interpretation of vector equations 
in quaternions may be readily deduced by the reader. 

120.] The following transformation is enuntiated without proof 
by Hamilton {Lectures, p. 587, and Elements, p. 299). 

»--i(rY)*5-i = U{rq-\-KrKq). 

To prove it, let r~\r^g^)^g~^ = t, then 

Tt = 1, and therefore 

But {r^ff = rti, 

or r'^q^ = rtqrtq^, 

or rq^ = tgrt. 

Hence KqKr - t-'^KrKqr\ 

or KrKq = tKqKH. 

Thus we have jji^^^ + ^^^^j = tU{qr±KqKr) t, 
or, if we put * = U{qr + KqKr), 

Ks= ± Ut. 
Hence sKs = {Tsf = 1 = ± stst, 

which, if we take the positive sign, requires 

st= ±\, 
or t= +«-!= ±UKs, 

which is the required transformation. 

[It is to be noticed that there are other results which might 
have been arrived at by using the negative sign above ; some in- 
volving an arbitrary unit- vector, others involving the imaginary of 
ordinary algebra.J 

121.] As a final example, we take a transformation of Hamil- 
ton's, of great importance in the theory of surfaces of the second 


Transform the expression • 

in which a, 13, y are any three mutually rectangular vectors, into 

the form mt , \ 2 

MW + PkV ^ 

which involves only two vector-constants, t, k. 

{T{ip + pK)}^ = {tp+pK){pi + Kp) (§§ 52, 55) 

= (l^ 4- K2)p2 + (tpKp +p/Cpt) 
= {l^+K^)p^+2S.lpKp 
= {l-K)Y+4:SipSKp. 

Hence (Sapf + iS^pf+iSypy^^^.p^ + i-^^'P^'P - 

(^2-12)2'' ' ^(^2-12)2 

But a-2(5ap)2 + y3-2(<S'j3/))2 + y-2(;S'yp)2 = p2 (§§ 25, 73). 

Multiply by jB^ and subtract, we get 

The left side breaks up into two real factors if ^2 be intermediate 
in value to a^ and y^ : and that the right side may do so the term 
in p2 must vanish. This condition gives 

ft— k)2 
fl2 = A^ L^ ■ and the identity becomes 

^(aV(l-5) + yV(^-l))p^(aV(l-5)^-yV(^-l)> = 4^. 
Hence we must have 

lL^^=^(a^il-.^) + yV{^-l)}, 

where ^ is an undetermined scalar. 

To determine j9, substitute in the expression for p^, and we find 

= {P^ + -^)(a^-7')-2(tt^+y^) t4|32. 

64 QUATERNIONS. [l22. 

Thus the transformation succeeds if 
1 2(a'= + /) 

1 / o? 

which gives jo+ - = + 2^/ 2_^ ' 

1 v^ 

Hence J^^ = (jj-y) (a^-/) = ± 4^/^^ 

or (/c2-i2)-i= iTayy. 

. . Ta^Ty 1 2'a-yy 

^^^"^' ^ = 77=^' 'P=W^^' 

and therefore 

ro-I-y , , ^2_a2 y2_^2 

Thus we have proved the possibility of the transformation, and 
determined the transforming vectors i, k. 
123.] By diflFerentiating the equation 

we obtain, as will be seen in Chapter IV, the following, 

where p also may be any vector whatever. 

This is another very important formula of transformation ; and 
it will be a good exercise for the student to prove its truth by 
processes analogous to those in last section. We may merely 
observe, what indeed is obvious, that by putting p'= p it becomes 
the formula of last section. And we see that we may write, with 
the recent values of i and k in terms of a, /3, y, the identity 

aSap + l3S^p + ySyp = ^ \J_,2yL 

_ (t — Kfp + 2 {iSkp + kSip) 

123.] In various quaternion investigations, especially in such 
as involve imaginary intersections of curves and surfaces, the old 
imaginarj' of algebra of course appears. But it is to be particularly 


noticed that this expression is analogous to a scalar and not to a 
veetorj and that like real scalars it is commutative in multiplica- 
tion with all other factors. Thus it appears, by the same proof as 
in algebra, that any quaternion expression which contains this 
imaginary can always be broken up into the sum of two parts, one 
realj the other multiplied by the first power of v— 1. Such an 
expression, viz. ? = /+ V^?", 

where ({ and /' are real quaternions, is called a biquaternion. 
Some little care is requisite in the management of these expressions, 
but there is no new diflBeulty. The points to be observed are : first, 
that any biquaternion can be divided into a real and an imaginary 
part, the latter being the product of \/— 1 by a real quaternion ; 
secondj that this \/ — 1 is commutative with all other quantities in 
multiplication ; tbirdj that if two biquaternions be equal, as 

we have, as in algebra, /= /, j"= /'j 

so that an equation between biquaternions involves in general eight 

equations between scalars. Compare § 80. 

124.] We have, obviously, since ^/— i is a scalar, 

Hence (§103) 

= {8q'+^^-i.Sf+ ?Y+ ^/irTr/')('S/+ V^^/'- rq'- sf- 1 Yf) 
= (Sq'+ ,y^I\Sff-{rq'+ ^/^^/')^ 
= {Tq'f - {Tff + 2 aA^aS. ^Kf. 
The only remark which need be made on such formulae is this, that 
the tensor of a hiquaternion may vanish while both of the component 
quaternions are finite. 

Thus, if ^/= Tq", 

and S.q'Kq"= 0, 

the above formula gives 

The condition S.^Kq"= 

may be written 

Kq"=q'-^a, or q"= -aKq'-^=- ^^r 
where a is any vector whatever. 

6 6 QUATERNIONS. [ 1 2 5 . 

Henee Tq' = Tq" = TKq" = ^ , 

and therefore 

Tq\Uq'- </::::\Ua.U^) = (l - ^/^^Ua)^ 

is the general form of a biquaternion whose tensor is zero. 

125.] More generally we have, q, r, ^, / being any four real and 
non-evanescent quaternions, 

{qJr '/^cf) (r+ ^/^T/) = qr-c['/+ ^^Ix^q/ Jf^r). 
That this product may vanish we must have 

qr = q'/,- 
and q/= —q'r. 

Eliminating / w;e have qq'~^qr = — /?', 
which gives {l'~^s)^ = ~^> 

i.e. q = ({a 

where a is some unit-vector. 

And the two equations now agree in giving 
— r = a/, 
so that we have the biquaternion factors in the form 
/(a+V^) and — (a-^/^)/; 
and their product is 

-/(a+ ^T-i) (a- sT^y, 
which, of course, vanishes. 

[A somewhat simpler investigation of the same proposition may 
be obtained by writing the biquaternions as 

g^C^-^^+y^) and (?-/-i+^/3i)/, 
or g'(/'+V^) and (Z'+v'ZIT)/, 
and shewing that 

5"= — /'= a, where Ta = 1.] 
From this it appears that if the product of two biveciors 
p + trV — l and p' + ff'v— 1 

is zero, we must have 

^-ip = _pV-i = Ua, 

where a may be any vector whatever. But this result is still more 
easily obtained by means of a direct process. 

126.] It may be well to observe here (as we intend to avail our- 
selves of them in the succeeding Chapters) that certain abbreviated 


forms of expression may be used when they are not liable to confuse, 
or lead to error. Thus we may write 

T^q for {Tqf, 
just as we write ^os^fl for (eos Of, 

although the true meanings of these expressions are 

T{Ta) and cos (eos 0): 
The former is justifiable, as T{Ta) = Ta, and therefore T^d is not 
required to signify the second tensor (or tensor of the tensor) of a. 
But the trigonometrical usage is quite indefensible. 
Similarly we may write 

S^q for {Sqf, &c., 
but it may be advisable not to use 

as the equivalent of either of those just written ; inasmuch as it 
might be confounded with the (generally) different quantity 

S.q^ or S{q^), 
although this is rarely written without the point or the brackets. 

137.] The beginner may expect to be a little puzzled with the 
aspect of this notation at first ; but, as he learns more of the sub- 
ject, he will soon see clearly the distinction between such an ex- 
pression as S.FapriSy, 

where we may omit at pleasure either the point or the first F with- 
out altering the value, and the very different one 

which admits of no such changes, without altering its value. 

All these simplifications of notation are, in fact, merely examples 
of the transformations of quaternion expressions to which part of 
this Chapter has been devoted. Thus, to take 3. very simple ex- 
ample, we easily see that 

S.Va^r^y = SFapr^y = S.a^FjSy = SaF.^Ffiy = -SaF.{Ffiy)p 
= SaF.{Fy^)P = S.aF{yP)^ = S.F{yP)pa = SFy^F^a 

= S.y^F^a = &c., &c. 

The above group does not nearly exhaust the list of even the simpler 
ways of expressing the given quantity. We recommend it to the 
careful study of the reader. He will find it advisable, at first, to 
use stops and brackets pretty freely ; but will gradually learn to 
dispense with those which are not absolutely necessary to prevent 

F 2 



1. Investigate, by quaternions, the requisite formulse for changing 
from any one set of coordinate axes to another; and derive from 
your general result, and also from special investigations, the usual 
expressions for the following cases : — 

(a.) Rectangular axes turned abbut « through any angle. 

(b.) Rectangular axes turned into any new position by rota- 
tion about a line equally inclined to the three. 

(c. ) Rectangular turned to oblique, one of the new axes lying 
in each of the former coordinate planes. 

2. If Tp = Ta = T^ = 1, and S.a^p = 0, shew by direct transfor- 
mations that ^_ jj^p _ „^ j;r(p _^) ^ + ^in-SalB). 

Interpret this theorem geometrically. 

3. If Sa^ = 0, Ta=T^=l, shew that 

(1 +0™)^ = 2 cos^a^;8 = 2Sa^.a^^. 

4. Put in its simplest form the equation 

pS. Fa^ r^y Fya = aV. Fya Fafi + 6F. Fafi FjSy + c F. Ffiy Fya ; 
and shew that a = S.fiyp, &c. 

5. Prove the following theorems, and exhibit them as properties 
of determinants : — 

{a.) S.[a + ^){fi + y){y^a) = 2S.apy,- 

{h.) S.Fa^F^yFya = -(S.a^y)'^, 

(c.) S.F(a + l3)i^ + y)F{l3 + y){y + a)F{y + a){a+p) = -4(5.a/3y)^ 

(d.) S. F( Fafi Ffiy) F( Fj3y Fya) F{ Fya Fa^) = - {S.a^y)\ 

{e.) S.5€C = — \6{8.afiy)*, where 

b = F{Fia+l3){^ + y)F(l3 + y)(y + a)), 

t = F{FiP+y){y + a)F(y+a)(a + p)), 

{:=F{F(y + a)(^a + l3)F{a + l3)(l3 + y)). 

6. Prove the common formula for the product of two determi- 
nants of the third order in the form 

S.a^yS.a^^iyi^ — Saa^ <S/3aj Sya^ 
Safi, mi3i Syfi, 
Sayi Si3yi Syy^ 

7. If, in § 102, a, j8, y be three mutually perpendicular vectors, 
can anything be predicted as to Oi, jSj, yj ? If a, j3, y be rectangular 
unil vectors, what of Oj, p^, y^? 


8. If aj /3, y, a', 13', y be two sets of rectangular unit-vectors^ 
shew that Saa'= Syfi'SjSy'-S^fi'Syy', Sec, &c. 

9. The lines bisecting pairs of opposite sides of a quadrilateral 
are perpendicular to each other when the diagonals of the quadri- 
lateral are equal. 

10. Shew that 

(6.) S.q^=S^q-3SqT^rq, 
(e.) a^p^y^+S^al3y = r\afiy, 
(d.) S{r.a^yF.Pyar.yal3) = 4: Sa^S^ySyaS.a^y, 
(e.) r.q^= (3 S^q-T'' Vq) Yq, 
(/.) qVYq-^ = -Sq.Urq + TFq; 
and interpret each as a formula in plane or spherical trigonometry. 

11. If g- be an undetermined quaternion, what loci are repre- 
sented by 

(a.) {qa-^r = -a^ 
{b.) {qa-^Y=a\ 
{e.) S.{q-aY=a\ 
where a is any given scalar and a any given vector ? 

12. If ^ be any quaternion, shew that the equation 

is satisfied, not alone by Q,= ±q but also, by 

Q = + ^/~:^{Sq.JJVq-TYq). 

(Hamilton, Lectures, p. 673.) 

13. Wherein consists the difference between the two equations 

T^^=l, and (^^=-1? 
a ^a' 

What is the full interpretation of each, a being a given, and p an 
undetermined, vector ? 

14. Find the full consequences of each of the following groups of 
equations, both as regards the unknown vector p and the given 

vectors a, /3,y:— „ „ on 

Sap = 0, Sap = 0, 

(«•) of'' " I' (*•) ^•''^P = '^' ^'-^ ^•"^'' = ^' 

S.pyp = 0; g^^ ^Q. S.a^yp = 0. 

15. From §§ 74, 109, shew that, if e be any unit-vector, and m 
any scalar, c" = cos — + e sm — • 


Hence shew that if a, j3, y be radii drawn to the corners of a tri- 
angle on the unit-sphere, whose spherical excess is m right angles, 

/3 + y'a+/3'y + a 
Also that, if A, B, C be the angles of the triangle, we have 

i£ iB iA 

y" ^"a" = — 1. 

16. Shew that for any three vectors o, j3, y we have 
{Ua^)^ + {UpY)'^+{Uayy + {U.a^yy + iUay.SUa^SUpy = -2. 

(Hamilton, Elements, p. 388.) 

17. If «i, Og, ag, OS be any four scalars, and p-^, p^, pg any three 
vectors, shew that 

+ 2n(aj2 + Spyp^ + ttjO,^ = 2n(a!2 -f p^) + 2n«'' 
+ 2{(a!2 +%'' +Pi^) ((^p^pg)'' + 2 «A(aj2 + Sp^pg) -a!^(p2-ps)^)} ; 
where Yla^ = a^a^a^. 
Verify this formula by a simple process in the particular case 

«j = 02 = 03 = a; = 0. 




128.] In Chapter I we have already considered as a special case 
the differentiation of a vector function of a scalar independent 
variable: and it is easy to see at once that a similar process is 
applicable to a quaternion function of a scalar independent variable. 
The differential, or differential coefficient, thus found, is in general 
another function of the same scalar variable ; and can therefore be 
differentiated anew by a second, third, &c. application of the same 
process. And precisely similar remarks apply to partial differentia- 
tion of a quaternion function of any number of scalar independent 
variables. In fact, this process is identical with ordinary differ- 

129.] But when we come to differentiate a function of a vector, 
or of a quaternion, some caution is requisite ; there is, in general, 
nothing which can be called a differential coefficient ; and in fact 
we require (as already hinted in § 33) to employ a definition of a 
differential, somewhat different from the ordinary one but, coinciding 
with it when applied to functions of mere scalar variables. 

130.] If r=F{q) be a function of a quaternion q, 

d^ = dFq = ^^n {F{q + '^±)-F{q)}, 

where » is a scalar which is ultimately to be made infinite, is defined 
to be the differential of r or Fq. 

Here dq may be any quaternion whatever, and the right-hand 
member may be written /., g s 

where / is a new function, depending on the form of F; homo- 
geneous and of the fi,rst degree in dq ; but not, in general, capable 
of being put in the form f ^^) j^_ 

7 2 QUATERNIONS. [ 1 3 1 . 

131.] To make more clear these last remarks, we may observe 
that the function y/„ g^ 

thus derived as the differential of V{q), is distributive with respect 
to dq. That is y (^^ ^ + ,) = y (^, ^) + y (^, ,)^ 

r and « being any quaternions. 

For /(?, r + *) = ^^ « (i? (^ + ^) - i^-C?)) 

And, as a particular case, it is obvious that if a; be any scalar 

/fe <»r) = isfiq, r). 

132.] And if we define in the same way 

dF{q,r,s ) 

as being the value of 

■C.«|'(s+*' '+*••+*• )-^(^.'.'. )}■ 

where q,r,Sy... dq, dr, ds, are any quaternions whatever ; we 

shall obviously arrive at a result which may be written 

f{q, r, s, ...dq, dr, ds, ), 

where ./ is homogeneous and linear in the system of quaternions 

dq, dr,ds, and distributive vrith respect to each of them. Thus, 

in differentiating any power, product, &c. of one or more quater- 
nions, each factor is to be differentiated as if it alone were variable ; 
and the terms corresponding to these are to be added for the com- 
plete differential. This differs from the ordinary process of scalar 
differentiation solely in the fact that, on account of the non-com- 
mutative property of quaternion multiplication, each factor must in 
general be differentiated in situ. Thus 

d{gr) = dq.r + qdr, but not generally = rdq + qdr. 
133.] As Examples we take chiefly those which lead to results 
which will be of constant use to us in succeeding Chapters. Some 
of the work will be given at full length as an exercise in quaternion 

(1) {Tpf=-p^. 
The differential of the left-hand side is simply, since Tp is a scalar, 


1 3 3- J DltFEEENTIATION. 73 

That Of p^ is ^^n((^p + ±f -p^) 

= 2Spdp. 
Hence Tp dTp = -Spdp, 

or dTp=-S.Updp = sf'' 

dTp ^dp 
or -=i- = ;iS — 


(2) Again, p = TpUp 

dp = dTp.Up + TpdUp, 

, dp dTp dUp 

whence JL-iLj^i^ 

p Tp Up 

= .i + f by(.). 

Hence dUp _ -p-dp 

W~ J' 

This may be transformed into F-^ or -^-^ » &e. 

p2 Tp^ 

(3) iTqy = qKq 

2TqdTq = i(^X^) = ^^n^(q + ^J)K(q + ^) -qKq], 

= l.-(&±Mi^^,^Kdq), 

= qKdq + dqKq, 
= qKdq + K{qKdq) (§55), 
= iS.qKdq = iS.Kqdq. 
Hence dTq = S.UKqdq = S.Uq-'^dq 

since :Z^ = :?'% and 27X^ = ?7^-i. 

If 3' = p, a vector, Kq = Kp = —p, and the formula becomes 
dTp = —S. Updp, as in (1). 


dTq dq 



which gives 

dq dTq dUq 
q- Tq^ Uq' 

whence, as 


q Tq 

we have 

dq _ dUq 

f —^ 

i Uq 

74 QUATERNIONS. [134. 


(4) aif)=<..^(ii+^y-f) 

= qdq + dq.q 

= 2S.qdq + 2Sq.Fdq + 2Sdq.Vq. 
If g' be a vector, as p, Sq and Sdq vanisli, and we have 
d{p^) = 28pdp, as in (1). 

(5) Let q = r*. 
This gives dr^ = dq. But 

^ = d{q^) = qdq + dq.q. 
This, multiplied iy ^ and m^o Kq, gives 

and drKq = dq.Tq^+qdq.Kq. 

Adding, we have 

qdr + dr.Kq = {q^ + Tq^ + 2<%.j) <«j ; 

whence dq, i. e. <^^, is at once found in terms of dr. This process 
is given by Hamilton, Lectures, p. 628. 

(6) qq-^ = 1, 

qdq~^ + dq.q~^ = ; 
. • . dq-"^ = — q-^ dq.q-^. 
If gf is a vector, = p suppose, 

dq~^ = —p~^dp.p~^ 

p^ p p 

(7) q = Sq+Fq, 
dq= dSq + dFq. 

But dq = ^^j- + Fi:?^. 

Comparing, we have 

dSq = Sdq, dVq = Vdq. 
Since Xq = Sq— Vq, we find by a similar process 

<?X2 = Kdq. 
134.] Successive diflFerentiation of course presents no new dif- 

Thus, we have seen that 

d{q^) = dq.q + qdq. 


DiflFerentiating again, we have 

and so on for higher orders. 

If §' be a vector, as p, we have, §133(1), 
d{p^) = 2Spdp. 
Hence d^(p^) = 2{dpf + 2Spd^p, and so on. 

Similarly d^Up= -dA-Fpdp) • 

But d 

1 _ 2dTp 2Spdp 

Tp"^ ~ Tp^ ~ Tp*^ 
and d. Vpdp = V. pd^p. 
Hence -^^J^p =- ^(rpi,)^+ Wp ^ 2J^^^^ 

= - ^ ((^P^P)' +P' Fp^V- 2 Fp^p^p^p) * 

135.] If the first differential of q be considered as a constant 
quaternion, we have, of course, 

d^q = 0, d^q = 0, &e., 
and the preceding formulae become considerably simplified. 

Hamilton has shewn that in this case Taylor's Theorem admits of 
an easy extension to quaternions. That is, we may write 

f{q + xdq) =/{q) + xd/{q) + ~ d^iq) + 

if d'^q = ; subject, of course, to particular exceptions and limita- 
tions as in the ordinary applications to functions of scalar variables. 
Thus, let y($') = q^) and we have 

4f(q) = q^dq + qdq.q + dq.q^, 
d^/iq) = 2dq.qdq + 2q{dq)^ + 2idq)''q, 
d^f{q) = G{dq)\ 
and it is easy to verify by multiplication that we have rigorously 
(g- + xdqf= f + x{q^dq + qdq.q + dq.q^) + x" {dq.qdq 4 q {dqf + {dqfq) + a;^(dqf ; 
which is the value given by the application of the above form of 
Taylor's Theorem. 

As we shall not have occasion to employ this theorem, and as the 
demonstrations which have been found are all too laborious for an 
elementary treatise, we refer the reader to Hamilton's works, where 
he will find several of them. 

* This may be farther simplified ; but it may be well to caution the student that 
we cannot, for such a purpose, write the above expression as 

-^J.pidpYpdp + d'p.p-'- 2dpSpdp}. 

76 QUATERNIONS. [136. 

1 36.] To differentiate a function of a function of a quaternion 
we proceed as with scalar variables, attending to the peculiarities 
already pointed out. 

137.] A case of considerable importance in geometrical appli- 
cations of quaternions is the differentiation of a scalar function of p, 
the vector of any point in space. 

Let F{p) = C, 

where i^ is a scalar function and C an arbitrary constant, be the 
equation of a series of surfaces. Its differential, 

f{p, dp) = 0, 
is, of eourscj a scalar function : and, being homogeneous and linear 
in dp, § 130, may be thus written, 

Svdp = 0, 
where i; is a vector, in general a function of p. 

This vector, v, is easily seen to have the direction of the normal 
to the given surface at the extremity of p ; being, in fact, per- 
pendicular to every tangent line dp, §§ 36, 98. Its length, when F is 
a surface of the second degree, is as the reciprocal of the distance of 
the tangent-plane from the origin. And we will shew, later, that if 
p = ix+jy+&z, 

/ . d . d , d \ „ 


1 . Shew that 

(a.) d.SUq = s.Usr^=-s^Truq, 

(b.) d.rUq=r.Uq-^F^dq.q-^), 

(c.) d.TrUq = S^=:S^^SUq, 

{d.) d.a" = ^ a^+'^dm, 

(e.) d\Tq={^.dqq-^-S.{dqq-^f}Tq = -~Tqr^^' 

2. If Fp='2.Sap8l3p+iffp^ 

give dFp t= Svdp, 
shew that v = S T. ap^ + (^ + 2 Sa^) p. 



138.] We have seen that the differentiation of any function 
whatever of a quaternion^ q, leads to an equation of the form 

where/" is linear and homogeneous in dq^. To complete the process 
of differentiation, we must have the means of solving this equation 
so as to be able to exhibit directly the value of dq. 

This general equation is not of so much practical importance as 
the particular case in which dq is a . vector ; and, besides, as we 
proceed to shew, the solution of the general question may easily be 
made to depend upon that of the particular case j so that we shall 
commence with the latter. 

The most general expression for the function _/ is easily seen to be 
dr =/(§■, dq) = 2 V.adqh + S.cdq, 
where a, I, and c may be any quaternion functions of q whatever. 
Every possible term of a linear and homogeneous function is re- 
ducible to this form, as the reader may easily see by writing down 
all the forms he can devise. 

Taking the scalars of both sides, we have 

Sd^- = S.cdq = SdqSa + S.rdqFc. 
But we have also, by taking the vector parts, 

Fd?- = 2 r. adqb = Sdq.^ rab + -2,r.a{ Vdq) b. 

Eliminating Sdq between the equations for Sdr and Vdr it *is 
obvious that a linear and vector expression in Vdq will remain. 
Such an expression, so far as it contains Vdq, may always be reduced 
to the form of a sum of teims of the type aS.^Vdq, by the help of 
formula like those in §§ 90, 91. Solving this, we have Tdq, and 
Sdq is then found from the preceding equation. 

78 QUATERNIONS. [139. 

139.] The problem may now be stated thus. 
Find the value of p from the equation 

o5/3p+ai-S)3ip+ ... = 2.aSfip = y, 

where a, 13, a^, ^i, ...y are given vectors. [It will be shewn later 
that the most general form requires but three terms, i. e. six vector 
constants a, y3, a^, ^j, Og, /Sg in all.] 
If we write, with Hamilton, 

(j>p = 2.a<S)3p, 
the given equation may be written 

<pp = y. 

or p = (j>-^y, 

and the object of our investigation is to find the value of the in- 
verse function (jr'^, 

140.] We have seen that any vector whatever may be expressed 
in terms of any three non-coplanar vectors. Hence, we should ex- 
pect a priori that a vector such as <p(p4>p, or <j)^p, for instance, should 
be capable of expression in terms of p, <j)p, and (p^p. [This is, of 
course, on the supposition that p, (j)p, and (fi^p are not generally co- 
planar. But it may easily be seen to extend to this case also. For 
if these vectors be generally coplanar, so are <j)p, (p^p, and <j)^p, since 
they may be written <r, ifxr, and (/)V. And thus, of course, ^^p can 
be expressed as above. If in a particular case, we should have, for 
some definite vector p, <pp=gp where ^ is a scalar, we shall obviously 
have <^^p =g^p and ^^p =g^p, so that the equation will still subsist. 
And a similar explanation holds for the particular case when, for 
some definite value of p, the three vectors p, Kpp, <^^p are coplanar. 
For then we have an equation of the form 

^^p = Ap-i- Bijip, 
which gives (l>^p = A(l>p + £(l)^p 

= ABp-\-{A + B^)<i>p. 
So that (p^p is in the same plane.] 
If, then, we write 

-(t,^p = xp+y4>p + e(l)^p, (1) 

it is evident that x, y, z are quantities independent of the vector p, 
and we can determine them at once by processes such as those iu 

If any three vectors, as «', /, h, be substituted for p, they will in 
general enable us to assign the values of the three coeflScients on 


the right side of the equation, andme solution is complete. For 
by putting (t>~^p for p and transposing, the equation becomes 

that is, the unknown inverse function is expressed in terms of direct 
operations. If x vanish, while y remains finite, we substitute ^~V 
for p, and have -y (^-^ = «p + cj,p, 

and if x and _y both -vanish 

— Z(j>~^p = p. 

141.] To illustrate this process by a simple example we shall 
take the very important case in which <f) belongs to a central surface 
of the second order ; suppose an ellipsoid ; in which case it will be 
shewn (in Chap. VIII.) that we may write 

^p = —a^iSip — h^jSjp—c^JcSkp, 
Here we have 

ipi = cp'i, <^H = aH, <f)H = a^i, 

4,j = by, <t>y = b*j, <t>y = by, 

(pk = cH, ^H = c*/i, ^^k = o^h. 
Hence, putting separately i,j, Tc for p in the equation (1) of last 
section, we have —a^ = x^ya^-\-m^, 

—b^= le+yb^+zb*, 

— C® = X-\-i/C^ +ZC*. 

Hence a^, b^, c^ are the roots of the cubic 
^* + «P +.?'£+«= 0, 
which involves the conditions 

z=-{a^ + l^ + c^), 
y = cfib"^ + b'^c^ + c^a^, 
x = — a^b^c^. 

Thus, with the above value of ^, we have 

(/>3p = aWc^p - {aW + h^c^ + c V) # + {a^ + b^-\- c^) <p^p. 
142.] Putting ^"^(T in place of p (which is any vector whatever) 
and changing the order of the terms, we have the desired inversion 
of the function ^ in the form 

aWc^-'^a- = {aW + bH^ + (^a^) a—{a'^ + b^ + c^) (fxr + ^V, 
where the inverse function is expressed in terms of the direct func- 
tion. For this particular case the solution we have given is com- 
plete, and satisfactory; and it has the advantage of preparing the 
reader to expect a similar form of solution in more complex cases. 

80 QUATERNIONS. [143. 

143.] It may also be useful as a preparation for what follows, if 
we put the equation of § 141 in the form 
= *(p') = 4,^p-{a^ + 6^ + c^)(l>''p + {aH'^ +¥c^ +c^a^)^p-a%^c^ p 

= {(«^-«') (</>-*') (<^-«')}p (2) 

This last transformation is permitted because </> is commutative with 
scalars like a*, i. e. <p{a^p) = a^^p. 

Here we remark that (by § 140) the equation 
r.p0p = 0, or ^p = gp, 
where g is some undetermined scalar, is satisfied, not merely by 
every vector of null-length, but by the definite system of three rect- 
angular vectors Ai, Bj, Ck whatever be their tensors, the corre- 
sponding particular values of g being a^, h^, c^. 

144.J We now give Hamilton's admirable investigation. 
The most general form of a linear and vector function of a vector 
may of course be written as 

</)p = 'S.V.qpr, 
where q and r are any constant quaternions, either or both of which 
may degrade to a scalar or a vector. 

Hence, operating by S.a- where o- is any vector whatever, 

S(r(l>p = 2ScTF.qpr = '28pF.raq = 8p4)'(T, (3) 

if we agree to write ^'o- = IiF.raq, 

and remember the proposition of § 88. The functions <^ and <j/ are 
thus conjugate to one another, and on this property the whole in- 
vestigation depends. 

145.] Let A, p. be any two vectors, such that 
^p ^ Vkp,. 
Operating by SX and S.p. we have 

8k<^p = 0, Sp.(t>p = 0. 
But, introducing the conjugate function <^', these become 

Sp(f>'K = 0, Sp^'p. = 0, 
and give p in the form mp = Fcjt'kcli'p,, 

where mis a scalar which, as we shall presently see, is independent 
of A, jM, and p. 

But our original assumption gives 

p = <(>-W\ix; 

hence we have m^~Wkp. = F^'k(p' p., (4) 

and the problem of inverting <^ is solved. 




146,] It remains to find the value of the constant m, and to 
express the vector Vd/kcb'u 

as a function of FX/n. 

Operate on (4) by /S.^'r, where v is any vector not coplanar with 
X and /n, and we get 

mS.(j/v(l>-^F\n = mS.v<i><irWKix (by (3) of § 144) 
= mS.Kixv = S.^'X^'ji^'v, or 

m = 



J3 q 



Pi ix 


H 2'2 


p q 



Pi Si 


[That this quantity is independent of the particular vectors \, ju, v 
is evident froija the fact that if 

k'=p)\. + qiJL + ri>, i/ = pjk + q-i^ix + r.^v, and d'= j?2^+S'2M+»"2»' 
be any other three vectors (which is possible since X, [x, v are not 
coplanar), we have 

<i)'k'= p<i)KJrq^' !!.-{- r<l>'v, &C., &C.-, 
from which we deduce 


so that the numerator and denominator of the fraction which ex- 
presses m are altered in the same ratio. Each of these quantities 
is in fact an Invariant, and the numerical multiplier is the same for 
both when we pass from any one set of three vectors to another. 
A still simpler proof is obtained at once by writing A +j3/x for \ 
in (5), and noticing that neither numerator nor denominator is 

147.] Let us now change ^ to <i>-\-g, where g is any scalar. It 
is evident that ^' becomes <i>'+g, and our equation (4) becomes 
mg{4>^-g)-WkiJ,= r{4,''+g)k{<t>'+g)ixi 

= r<t>'k^'ix+gF((l/k,j. + k<t>',x)+g'rk^, 
= {'m(t)~^ +gx+g^)V^kix suppose. 
In the above equation 

_ S.{cl>',+g)k{ct/+g)t,{^'+g)v 

'^'- sJili, 

= m+m^g+m^g^+g^ 

82 QUATERNIONS. [148. 

is what m becomes when ^ is changed into ^-Vg; % and m^ being 
two new scalar constants whose values are 

"^ Sl^v ' 

_ S. {kij.(f>'v + 4>' kfiv + X.(l>'iJLv) 

If, in these expressions, we put k+pjx for \, we find that the terms 
in jp vanish identically ; so that they also are invariants. Substi- 
tuting for Mg, and equating the coefficients of the various powers 
of ^ after operating on both sides by ^-f-^, we have two identities 
and the following two equations, 

% = '^ + X. 

[The first determines x, and shews that we were justified in treat- 
ing F{((/\ij,-\-\<f>'^i) as a linear and vector function of F.Xi/,. The 
result might have been also obtained thus, 

SAx^Xfi. = S.\<f/\ix=—S.\ix(t/\=-8.\(l)rhiJ., 
S.fjLx^^fJ^ = S.jjlKcj/ij, = —S.iiipVKjj,, 
S.vxVXix. = S.{v^'Xii. + vk4>'i).) 
= m2SKij,v—S.\iJi.^'v 
= S.v {m^Vkfi—^fKii) ; 
and all three (the utmost generality) are satisfied by 

X = %- *-J 
148.] Eliminating ^ from these equations we find 

or m<l)~^ = OTj — ^j (^ 4- (/)^, 
which contains the complete solution of linear and vector equations. 

149.] More to satisfy the student of the validity of the above 
investigation, about whose logic he may at first feel some diffi- 
culties, than to obtain easy solutions, we take a few very simple 
examples to begin with : we treat them with all desirable prolixity, 
and we append for comparison easy solutions obtained by methods 
specially adapted to each case. 

150.] Example I. 

Let <l>p = V.apfi = y. 

Then <^'p = V.^pa = <^p. 

Hence m = -=r^ — S ( V. aX^ V. au/3 V. av^). 

8.\iJ,v ^ ' 


Now X, n, V are any three non-eoplanar vectors; and we liiay 
therefore put for them a, ^,y if the latter be non-coplanar. 
With this proviso 

% = -s-^'Sf(a2/3.a/3'2.y + a.a/32.r.oy^ + a2/3.j3.r,ay/3) 


= — O' 

= —Sap. 

S (Ti^^.yS.y + a.a/32.y + o;8 V.ayfi) 


which is one form of solution. 

By expanding the vectors of products we may easily reduce it to 

the form a^^^Safi.p = - a^/S^ y + a^^Say + Ba^Sfiy, 

a-^Say + B-^S3y—y 
or p = — ' ■ — -^ — - ■ 

151.] To verify this solution, we have 

^•"''^ "= ^O-^ay + a-^/Sy-r.ay/S) = y, 
which is the given equation. 

153.] An easier mode of arriving at the same solution, in this 
simple case, is as follows : — 

Operating by S.a and S.p on the given equation 
r.opjS = y, 
we obtain a^SjSp = Say, 

P^Sap = S^y ; 
and therefore aSfip = a~'^Say, 

pSap = /8-i/S'/3y. 
But the given equation may be written 

aS^p—pSa^ + ISSap = y. 
Substituting and transposing we get 

pSafi = a'^Say + p-^S^y—y, 
which agrees with the result of § 150. 

153.] If a, fi, y be coplanar, the above mode of solution is appli- 
cable, but the result may be deduced much more simply. 

For (§101) S.aPy = 0, and the equation then gives S.a^p = 0, so 
that p is also coplanar with a, /3, y. 

6 3, 

84 QUATEENIONS. [154. 

and at once „ _ „-i,,o-i 

Hence the equation may be written 

app = y, 

P = a"V^' 
and this, being a vector, may be written 

This formula is equivalent to that just given, but not equal to it 
term by term. [The student will find it a good exercise to prove 
directly that, if o, /3, y are coplanar, we have 

^(a-i/Sfay + ^-i*S/3y-y) = a-'^S^'^y^^-^Sar'^y-ySar'^^-'^r^ 

The conclusion that o a n 
b.app = 0, 

in this case, is not necessarily true if 

5a/3 = 0. 

But then the original equation becomes 

aS^p + pSap =: y, 

which is consistent with 

S.aPy = 0. 
This equation gives 

^("'^-^«^) = «U/y ^A + ^ Say S ' 
by comparison of which with the given equation we find 

Sap and S^p. 
The value of p remains therefore with one indeterminate scalar. 
154. J Example II. 
Let <^p =: V.a^p = y. 

Suppose a, ;8, y not to be coplanar, and employ them as A, ft, v to 
calculate the coefficients in the equation for (j)"^. We have 

S.(T(j)p = S.cra^p = S.pKcra^ = S-pcj/a: 
Hence <^'p = ^-palS = V.I3ap. 

We have now 

= a^fi^Safi, 

m, = -=-— -(5(0.^0/3. r, /Say + ;3a2./3.r.;3ay + ;3a2./3a/3.y) 

= 2{Safif + a^^^, 

«*2 = "cV S(a.^.r.fiay+a.^a^.y + ^aK^.y) 

= 38al3. 



= (2 (<So^)2 + a^j3^) y- 3/Sa/3 V.a^y + V.a^ V.a^y, 
which, by expanding the vectors of products, takes easily the simpler 
form a^p2Sa^p ^ a^^2y_^^2s^^ ^ 2^Sa^Say-^a?S^y. 

155.] To verify this, operate by F.a/3 on both sides, and we have 
a^^^Sapr.aPp = a'^^W.afiy- r.a^afi'^Say+2ap^Safi8ay-ao?^'^S^y 
= a?^^ {a8^y-pSay + ySaP)-{2aSap-^a^)P^Say 

+ 2 afi^Sa^Say—aa^^^S^y 
= a^p^Sa^.y, 

or V. afip = y. 

156.] To solve the same equation without employing the general 
method, we may proceed as follows : — 

y = r. a^p = pSa^ + V. r{a^) p, 
Operating by S. Fa^ we have 

S.a^y — S.a^pSa^. 
Divide this by Sa^, and add it to the given equation. We thus 

obtain o o 

y + ^^ =pSal3+ r. Viafi) p + S. r{afi) p, 

= {Sal3+ral3)p, 
= a/3p. 

Hence p = /3-1 a-i (y + -^) , 

a form of solution somewhat simpler than that before obtained. 

To shew that they agree, however, let us multiply by a^^^Safi, 
and we get a^/i^Sa^.p = ^aySa^ + fiaS.a^y. 

In this form we see at once that the right-hand side is a vector, 
since its scalar is evidently zero (§ 89). Hence we may write 

a?^^Sa^.p = r.^aySa^-Va^S.a^y. 
But by (3) of §91, 

—yS.ap ra/3 + a/S./S ( Fa^) y + /3/S. F{aP) ay + Fa^S.a^y = 0. 
Add this to the right-hand side, and we have 
a^P^Sa^.p = y {{Sa^)^-S.al3Fap)-a {Sa0S^y-^S.^ (Fafi) y) 

+ ;8 {Sal3Say + S.F (afi) ay) . 
But {Safif-S-a^Fa^ = {Sa^f-{Fa^f = a^^\ 

Sa^8^y-S.fi{Fa^) y = Sa^Sfiy-SftaS^y + ^^Say = ^^Say 
SapSay + S.F{aP)ay = SafiSay + SafiSay-a^S^y 
= 2Sa^Say-a'^S^y; 
and the substitution of these values renders our equation identical 
with that of § 154. 

86 QUATERNIONS. [157. 

[If n, /3, y be coplanar, the simplified forms of the expression for p 
lead to the equation 

Sap.p-^a-^y = y-a-^Say + 2pSa-^fi-^Say-p-'8l3Y, 
which, as before, we leave as an exercise to the student.J 

157.] Example III. The solution of the equation 

Tep = y 
leads to the vanishing of some of the quantities m. Before, how- 
ever, treating it by the general method, we shall deduce its solution 
from that of V.a^p = y 

already given. Our reason for so doing is that we thus have an 
opportunity of shewing the nature of some of the cases in which one 
or more of m, m^, m^ vanish; and also of introducing an example 
of the use of vanishing fractions in quaternions. Far simpler solu- 
tions will be given in the following sections. 

The solution of the last- written equation is, § 154, 

a^^^Sa^.p = a^^^y-a^^Say—^a.'^S^y+2^Sa^8o.y. 
If we now put o^ = e + e 

where e is a scalar, the solution of the first-written equation will 
evidently be derived from that of the second by making e gradually 
tend to zero. 

We have, for this purpose, the following necessary transforma- 
tions : - a2^2 _ „^ x.a^ = (e + e) (e - e) = e^ - e^, 

a^^Say + ^a?8^y = a^.^Say + jSa.aS^y, 

= {e + e)fiSay + {e—e)aSPy, 
= e {^Say + aS^y) + eV.yVa^, 
= e l^Say + aS^y) + e Tye. 
Hence the solution becomes 

(e2_e2)ep = {e^-e.^)y-e{^Say + aS^y)-iryi + 2e^Say, 
- le^—(^)y + eF.yra^—eryf, 
= ^e^^i'')y + ery€ + yf'-fSyf, 
= e^y + eVye — tSye, 
Dividing by e, and then putting e = 0, we have 

-eV = rye-<„(^). 

Now, by the form of the given equation, we see that 

Sye = 0. 
Hence the limit is indeterminate, and we may put for it cc, where as 
is any scalar. Our solution is, therefore, 

or, as it may be written, since Sye = 0, 
p = e-i(y + a;). 


The verification is obvious — for we have 

ep = y + x. 
158.] This suggests a very simple mode of solution. For we 
see that the given equation leaves Sip indeterminate. Assume, 
therefore, Sep = x 

and add to the given equation. We obtain 

ep =x + y, 
or p=e-i(y + a,), 
if, and only if, p satisfies the equation 

Vep = y. 
159.] To apply the general method, we may take e, y and ey 
(which is a vector) for A, p,, v. 
We find <l)'p = Vpe. 



m = 0, 


m„ = 0. 


That is. 


= e~^y + xe, as before. 

Our warrant for putting xe, as the equivalent of 0"^ is this : — 

The equation ^2^ _ q 

may be written r.eFfcr = = <re^ - tSta. 

Hence, unless o- = 0, we have o- 1| e = xe. 

160.] Example IV. As a final example let us take the most 
general form of t^, which, as will be soon proved, may be expressed 
as follows : — 

<^p = ajS/3p + a-^S^-^p + a^S^^p = y. 
Here <l>p = ^Sap + ^-^80^ p + ^.^Sa^ p, 

and, consequently, taking a, Oj , 02, which are in this case non- 
coplanar vectors, for A, p., v, we have 

-S.(^Saa + ^^Sa^a + P^Sa^oi) {l3Saa-^ + P-^Saj^aj^+ ...) {pSaa2+ ) 




Saa Sa-jO, iSogO 
Saor^ /SojOj (SogOj 
Saag /iSa^ag Sa^a^ 

= ^f^{ASaa + A^Sa^a + A,Sa^a), 


88 QUATERNIONS. [l6o. 

where A = Sa^aiSa^ai— Sai.aiSaia2 

= —S. VojO^ VojO^ 
A^ = Sa^aiSaa2—Saa,ySa^a3 

= —S. Fctg a FioiOg 
A2 = SacijSa^a2 — Saia^Saa^ 
r= — S. Vaa^ FajOg. 
Hence the value of the determinant is 
— {SaaS. FojO^ Va^a^ + SoyoS. Fa^a Va^a^ + Sa^aS. Faa^ FoiOg) 

= -SMiFaia^S.aaja^) {by § 92 (3)} = -{S.aa^a^f. 
The interpretation of this result in spherical trigonometry is very 
interesting, (^ee Ex. (6) p. 68.) 
By it we see that 


m, = g 8.[a (0Saai + ^,80.0, + /SgSoaai) {^Saa^ + ^i^ojOa + /SaiSaga,) + &c.] 


= 5 {S.a^Pi (800180102— 8010^8002) + ) 

= o (5.a/3i3i5.a r.Oi ragOi + ) 

= - ^ IS.a ( FpPy8. Faoi Foy02 + F^^8. F02O rajOg + Ffi^^2^. Fo^a^ Fa^a^ 

tf.aoia^ +S.ai{Fl3l3^8.FaoyFo20+ ) 

+ S.a2{Fl3Pi8.FaoyFooi+ )] ; 

or, taking the terms by columns instead of by rows, 

= — p 18. F^Pi {a8. Faoy Fa^a^ + 0^8. Faa^ Fa^ + 028. Faa-^ Faa-^ 

8.00^02 1 

= --^^^lS.Ffi^y{FoayS.oay02) + ], 

_ = -S{FaoiFpfii+ Fai02F^,p2+ T^V^^^i^)- 

«2„ = -r S\oaA^Saa2 + ^iSoia2+ ...) + a2o{^8oai-\- ...) + a-^02{pSaa+...)'\, 

or, grouping as before, 

= — 8\^{ Foai8ao2 + Fa.^o8aay + Foy028aa) 4- • ■ ■] , 


= -^^^S[fi {08.00^02)+ j (§92(4)), 

= 8(0^+0^^1+02^2)- 

And the solution is, therefore, 

(f>-^y8.aOya28.p^l^2 = pi^.aaiaa&^^i^a 

= y25. Fooy T/S/Sj + ^ySSo^S - <J) V- 


\ [It will be excellent practice for the student to work out in detail 
the blank portions of the above investigation, and also to prove 
directly that the value of p we have just found satisfies the given 

161. J But it is not necessary to go through such a long process 
to get the solution — though it wUl be advantageous to the student 
to read it carefully — ^for if we operate on the equation by S-OjCt^, 
S.a^a, and Smo^ we get 

S.aiOf^aSlSp = S.aiO^y, 
S.a^aoiSfiip =: S.a^ay, 
S.aayO^S^^p ^ S.aajy. 
From these, by § 92 (4), we have at once 

pS.aojO^S.pPiP^ = Fpi3^8.aajy+ r^^^S.a^a^y + V^^S.a^ay. 
The student will find it a useful exercise to prove that this is equi- 
valent to the solution in § 160. 

To verify the present solution we have 

= a'S'.;8/3ij32iS'.aj^a2y-|-aj^&;8j^^2/3'^-"2'*y + ''2'^-/32/3/5i'^-°"iy 
= S.^l3,fi^ {yS.aaia,), by § 91 (3). 

163.J It is evident, from these examples, that for special cases 
we can usually find modes of solution of the linear and vector equa- 
tion which are simpler in application than the general process of 
§ 148. The real value of that process however consists partly in 
its enabling us to express inverse functions of 0, such as ((^+^)~^ 
for instance, in terms of direct operations, a property which will be 
of great use to us later ; partly in its leading us to the fundamental 
cubic ^^—m^^^ +mj(f>—m = 0, 

which is an immediate deduction from the equation of § 148, and 
whose interpretation is of the utmost importance with reference to 
the axes of surfaces of the second order, principal axes of inertia, 
the analysis of strains in a distorted solid, and various similar 

163.] When the fiinction <^ is its own conjugate, that is, when 
Spcpo' = Scrcfip 
for all values of p and o-, the vectors for which 

{<t>-ff)p = 
form in general a real and definite rectangular system. This, of 
course, may in particular cases degrade into one definite vector, and 
any pair of others perpendicular to it ; and cases may occur in 
which the equation is satisfied for every vector. 

90 QUATERNIONS. [164, 

Suppose the roots oi mg= (§ 147) to be real and different, then 

VPi — ffiPi 1 ^j^ere p^, p^, P3 are three definite vectors determined 

Wi — y2P2 f jjy. ^jijg constants involved in <ji. 

<t>Pa =ffsP3' 

Hence ^i^2%P2 = '5-M#2 

= S.pT,4>^P2, or = S.p^ip^pi, 
because ^ is its own conjugate. 

But (^^2 = fflPz) 

and therefore 9x9i^P-iPi = 9l^PiP2 = ^fi^f 1P2 > 
which, as g^ and g^ are by hypothesis different, requires 

SP\P2 = 0- 

Similarly 'S/'2P3 = 0, 'S'pgpj^ = 0. 

If two roots be equal, as g^, g^, we still have, by the above proof, 
iSpiPg = and Sp^p^ = 0. But there is nothing farther to determine 
/>2 and P3, which are therefore any vectors perpendicular to py 

If all three roots be equal, every real vector satisfies the equation 

164.] Next, as to the reality of the three directions in this case. 

Suppose g^-'r^N — 1 to be a root, and let pg + tr^'v— 1 be the 
corresponding value of p, where g,^ and ^2 ai'c real numbers, pg and a^ 
real vectors, and v — 1 the old imaginary of algebra. 

Then ^{p^ + cTg a/— 1 ) = (^2 + >^2 ■v^^^) (P2 + "^a v'— ^). 
and this divides itself, as in algebra, into the two equations 

#2 = ^2^2 — '^2'^2) 
(/mJ-2= /?2P2+^2°'2- 

Operating on these by /S.o-g, /S.pg respectively, and subtracting the 
results, remembering our condition as to the nature of <^ 

<S'a-20P2 = Sp^^lT^, 

we have ^gC"'! +Pi) = 0. 

But, as o-g and p^ are both real vectors, the sum of their squares 
cannot vanish. Hence h^ vanishes, and with it the impossible part 
of the root. 

165. J When ^ is self-conjugate, we have shewn that the equa- 

^^^"^ g^— m^g^ + m-^g —m — Q 

has three real roots, in general different from one another. 
Hence the cubic in ^ may be written 

{.<i>-9i)i.^-9^{.4>-9s) = 0> 


and in this form we can easily see the meaning of the cubic. For, 
let pi, p2, pg be three vectorg such that 

{^-ffi)pi = 0. {'t>—ff2)P2 = 0, {<t>—9^Ps = 0. 
Then any vector p may be expressed by the equation 

pS-PiP2Pa = pAP2P3P + P2.^-P3PiP + Pa^-PiP2P (§91). 
and we see that when the complex operation, denoted by the left- 
hand member of the above symbolic equation, is performed on p, the 
first of the three factors makes the term in pj vanish, the second 
and third those in p^ and pg respectively. In other words, by the 
successive performance upon a vector of the operations <f> — ^j, (p — ff^' 
^—g^, it is deprived successively of its resolved parts in the direc- 
tions of Pi, p^, Pg respectively j and is thus necessarily reduced to 
zero, since pj, pg, pg are (because we have supposed g-^^, g^, g^ to be 
distinct) distinct and non-eoplanar vectors. 

166.] If we take pj, pg, pg as rectangular K^zi^- vectors, we have 
— p = p-iSpjp + P2,8p2P + Ps'SpaP, 
whence # = —SiPx^pT^P—g^p^Sp^p—g^^Sp^p ; 

or, still more simply, putting i, j, h for p^, pg, pg, we find that any 
self-conjugate function may be thus expressed 

^P = —9ii^P —ad^JP —9i^Skp, 
provided, of course, i, j, k be taken as roots of the equation 

Vp^p = 0. 
167.] A very important transformation of the self-conjugate 
linear and vector function is easily derived from this form. 

We have seen that it involves three scalar constants only, viz. y^, 
g , g^. Let us enquire, then, whether it can be reduced to the fol- 
lowing form <j)p =/p + AF.{i + e/i:)p{i— eh), 
which also involves but three scalar constants/, h, e. Here, again, 
i, y, h are the roots of Vp^p = 0. 
Substituting for p the equivalent 

p = —iSip—jSjp—kSkp, 
expanding, and equating coefficients of i,j, k in the two expressions 
for <\>p, we find —g^ = —/+ ^^ (2 — 1 -[- e% 

-g,=-.f-k{2e'^ + l-e^). 

These give at once 

-(^1-^2) = 2-^, 
-{9z-9z) = Se^/J. 

92 QUATERNIONS. [l68. 

Hence, as we suppose the transformation to be real, and therefore e^ 
to be positive, it is evident that ffi — ff^ and ^2 — ffa have the same 
sig^ ; so that we must choose as auxiliary vectors in the last term 
of <pp those two of the rectangular directions i, j\ k for which the 
coefficients g have the greatest and least values. 
We have then ^i^9j-h., 

^=-\ {91-92), 
and f=\{gi+gi). 
168.] We may, therefore, always determine definitely the vec- 
tors \, fi, and the scalar y, in the equation 

when <\> is self-conjugate, and the corresponding cubic has not equal 
roots, subject to the single restriction that 

is known, but not the separate tensors of X and fx. This result is 
important in the theory of surfaces of the second order, and will be 
considered in Chapter VII. 

169.] Another important transformation of <^ when self-conju- 
gate is the following, ^p = aaVap + i^Sfip, 

where a and b are scalar s, and a and /3 unit-vectors. This, of 
course, involves sis scalar constants, and belongs to the most gen- 
eral form 4)p = —giPiSpiP—g2P2^P2p—9aP3^PaPy 
where pi, pg, p^ are the rectangular unit-vectors for which p and (pp 
are parallel. We merely mention this form in passing, as it be- 
Ipngs to the Jveal transformation of the equation of surfaces of the 
second order, which will not be farther alluded to in this work. It 
will be a good exercise for the student to determine a, ;8, a and b, 
in terms of i^^, yg. 93, ai"i Pi> P2, Pa- 

170.] We cannot afford space for a detailed account of the sin- 
gular properties of these vector functions, and will therefore content 
ourselves with the enuntiation and proof of one or two of the most 

In the equation nKp'^FXpi. = F(\>'\<\)'ii. (§ 145), 
substitute \ for ^'K and ji for <^'p., and we have 

»«rqb'-i\<^'-V = ^FKix. 
Change ^ to <p+g, and therefore ^' to <\> +g, and m to %, we have 

a formula which will be found to be of considerable use. 


171.] Again, by § 147, 

Similarly -^ S.p (</> + ^)- V = t ■^P^" V + ^P + ^P^- 

'^S.pi^+,)-^p-^S.pi^ + A)-^P = i,-,)[p^-'^]. 

That is, the functions 

are identiealj i. e. when equated to constants represent the same series 
of surfaces, not merely when 

g = h, 
but also, whatever be g and h, if they be scalar functions of p which 
satisfy the equation mS.p(j>-^p = gkp'^. 

This is a generalization, due to Hamilton, of a singular result ob- 
tained by the author *. 
173.] The equations 

S.p((l>+g)-^p = 0,l 

S.p{<p + A)-^p=0,i ^ ' 

are equivalent to mSp(j)~^p+gSp\p+ff^p^ = 0, 
mSp<t>-'^p + hSpxp + h^p^ — 0. 
Hence m{\—x) Sp4>-''-p + {g—M) Spxp + {g^ — A^(c)p^ = 0, 
whatever scalar be represented by x. 

That is, the two equations (1) represent the same surface if this 
identity be satisfied. As particular cases let 

(1) aj=l, in which case 

Sp-\p+g + h = 0. 

(2) g—hx=0, in which case 

m{l - |)^p-i0-V + (/->^^f) = 0, 
or mSp~^4>~^P~9^ — "• 

(3) a- = |a. giving 

m{\- |-,) -Spr V + (^ - >^ fg) *PXP = 0, 

or m {A+g)Sp(l>-^p +g/iSpxp = 0. 

* Note on the CarteBian equation of the Wave-Surfaee. Quarterly Math. Jowmal, 
Oct. 1859. 

94 QUATERNIONS. [l73- 

173.] In various investigations we meet with the quaternion 
J = a<l>a + I3<j>p + y<l>y, 
where a, /3, y are three unit-vectors at right angles to each other. 
It admits of being put in a very simple form, which is occasionally 
of considerable importance. 

We have, obviously, by the properties of a rectangular unit- 
system ^ _ ^y^a + yatl>l3 + a^<}>y. 
As we have also s.afiy = _ i (§71(13)), 
a glance at the formulae of § 147 shews that 

at least if ^ be self-conjugate. Even if it be not, still (as will be 
shewn in § 1 74) ^p = ^'p + r^p, 

and the new term disappears in Sq. 
We have also, by § 90 (2), 

Vq=a{Sfi(l>y-Sy<pp)-i-P{Sy<j>a-Sa4r/) + y{Sa(l>^-Sl3<f>a) 
= a8fi{4)~<l/)y + fiSy{(t>-<tt)a + ySa{(l>—^')^ 
= aS.fiey + ^S.yea+ yS.aep 
= — {aSae + /S/S/Se + ySye) = e. 
[We may note in passing that this quaternion admits of being 
expressed in the remarkable form 

where V = OT-+S-7-+y-5-> 

ax ay dz 

and p=ax-\-^y-\-yz. 

We will recur to this towards the end of the work.] 

Many similar singular properties of <\> in connection with a rect- 
angular system might easily be given ; for instance, 
V{a F<l>^(j>y + j3 Ffycl)a + y F^a<p^) 

= mF(a<j)-^a + fi^'~^fi + y^'-'^y) = mF.Vi^'-'^p = 4>e ; 
which the reader may easily verify by a process similar to that just 
given, or (more directly) by the help of § 145 (4). A few others 
will be found among the Examples appended to this Chapter. 

174.] To conclude, we may remark that as in many of the 
immediately preceding investigations we have supposed <f> to be 
self-conjugate, a very simple step enables us to pass from this to 
the non-conjugate form. 

For, if ^' be conjugate to (^, we have 
Sp(j>'(r = 8<T<pp, 
and also Spt^xr = Sa-^'p. 

17 7-] soLUTioi^r OF equations. 95 

Adding, we have 

SO that the function {<f> + <j)') is self-conjugate. 

Again, Sp(f>p = Spcj/p, 

which gives Sp{^—<^')p =. 0. 

Hence {<f>-~<l)')p = Fep, 

where, if ^ be not self-conjugate, e is some real vector, and therefore 
<t>P = \{<l> + <l>')p+\{4>-^')p 

=U<P + <t>')p+hrip. 

Thus every non-conjugate linear and vector function differs from 
a conjugate function solely iy a term of the form 

The geometric signification of this will be found in the Chapter on 

175.] We have shewn, at some length, how a linear and vector 
equation containing an unknown vector is to be solved in the most 
general case ; and this, by § 138, shews how to find an unknown 
quaternion from any sufficiently general linear equation containing 
it. That such an equation may be sufficiently general it must have 
both scalar and vector parts : the first gives one, and the second 
three, scalar equations ; and these are required to determine com- 
pletely the four scalar elements of the unknown quaternion. 

176.] Thus Tq = a 

being but one scalar equation, gives 

q = aJJr, 
where r is any quaternion whatever. 

Similarly Sq — a 

gives q — a +6, 

where d is any vector whatever. In each of these eases, only one 
scalar condition being given, the solution contains three scalar in- 
determinates. A similar remark applies to the following : 

Trq = a 
gives q = x + ad, 

and SUq = cos a, 

gives q = 006'^ , 

in each of which x is any scalar, and any unit vector. 
177.] Again, the reader may easily prove that 

^'^ QUATEENIONS. [178. 

where a is a given vector, gives, by putting Sq = x, 

Faq = p + cca. 
Hence, assuming Saq = y, 

we have aq=y + iDa + p, 

or ? = «+yo-i + a-^j8. 

Hercj the given equation being equivalent to two scalar con- 
ditions, the solution contains two scalar indeterminates. 

178.] Next take the equation 
Faq = p. 
Operating by 8.a-\ we get 

Sq = 8a-^fi, 
so that the given equation becomes 

ra{Sa-^p+rq) = p, 
or FaFq = ^-aSa-^fi = aVa'^ 

From this, by § 168, we see that 

rq = a-^{x + aVa-^fi), 
whence q = Sa-i/3 + a-^ {« + a Fa-i/S) 

= a-i(/3 + a!), 
and, the given equation being equivalent to three scalar conditions, 
but one undetermined scalar remains in the value of q. 

This solution might have been obtained at once, since our equation 
gives merely the vector of the quaternion aq, and leaves its scalar 

Hence, taking a; for the scalar, we have 
aq = Saq + Vaq 

179.] Finally, of course^ from 

0^ = 13, 
which is equivalent to four scalar equations, we obtain a definite 
value of the unknown quaternion in the form 

q = a-i^. 

180.] Before taking leave of linear equations, we may mention 
that Hamilton has shewn how to solve any linear equation con- 
taining an unknown quaternion, by a process analogous to that 
which he employed to determine an unknown vector from a linear 
and vector equation j and to which a large part of this Chapter has 
been devoted. Besides the increased complexity, the peculiar fea- 
ture disclosed by this beautiful discovery is that the symbolic 
equation for a linear quaternion function, corresponding to the cubic 


in (^ of § 162, is a biquadratic, so that the inverse function is .given 
in terms of the first, second, and third powers of the direct function. 
In an elementary work like the present the discussion of such a 
question would be out of place : although it is not very difficult to 
derive the more general result by an application of processes already 
explained. But it forms a curious example of the well-known fact 
that a biquadratic equation depends for its solution upon a cubic. 
The reader is therefore referred to the Mements of (Quaternions, 
p. 491. 

181.] The solution of the following frequently-occurring par- 
ticular form of linear quaternion equation 

aq + qb = c, 
where a, b, and c are any given quaternions, has been effected by 
Hamilton by an ingenious process, which was applied in § 133 (5) 
above to a simple case. 

Multiply the whole bi/ Ka, and into b, and we have 
T'^a.q + Ka.qb=Ka.c, 

and aqb-\-qb'^ = cb. 
Adding, we have 

q {T^a + b'^ + 2Sa.b) = Ka.c + cb, 

from which q is at once found. 

To this form any equation such as 

a'qh'+c'qd' = e' 
can of course be reduced, by multiplication by c'-^ and into b'"'^. 

183.] As another example^ let us find the differential of the cube 
root of a quaternion. If ^3 _ ,. 

we have q'^dq + qdq.q + dq.q^ = dr. 

Multiply by q, and into q~^, simultaneously, and we obtain 

q^dq.q~^ + q^dq + qdq.q = qdr.q-^. 
Subtracting this from the preceding equation we have 

dq.q^—q^dq.q~^= dr—qdr.q~^, 
or dq.q^—q^dq = dr.q—qdr, 

from which dq, or d{r^), can be found by the process of last section. 
The method here employed can be easily applied to find the 
differential of any root of a quaternion. 

183.] To shew some of the characteristic peculiarities in the 
solution even of quaternion equations of the first degree when they 
are not sufficiently general, let us take the very simple one 

aq = qb, 
and give every step of the solution, as practice in transformations. 


y** QUATERNIONS. [183. 

Apply Hamilton's process (§181), and we get 
T^a.q = Ka.gh, 
qh^ = aqb. 
These give q(THJrb'^-2bSa,) = 0, 

so that the equation gives no real finite value for q unless 

or b = Sa + l3TFa, 
where /3 is some unit-vector. 

By a similar process we may evidently shew that 

a = 8b + aTVb, 
a being another unit-vector. 

But, by the given equation, 

Ta = Tb, 
or S^a + T^ra = SH + TWb; 
from which, and the above values of «• and b, we sec that we may 
write So, Sb 

Wa = Wb=^' '^PP°''- 
If, then, we separate q into its scalar and vector parts, thus 
q = r + p, 
the given equation becomes 

{a. + a){r + p) = (r + p)(a + ^) (1) 

Multiplying out we have 

r{a—l3) = pfi — ap, 

which gives iS{a—p)p = 0, 

and therefore p = Fy{a—fi), 

where y is an undetermined vector. 
We have now 

r{a—p,) = p^-ap 

= ry{a-^).l3-aFy{a-p) 

= y{Safi+l)-{a-^)Spy + y{l+Sal3)-{a-fi)Say 

= -ia-l3)Sia + fi)y. 
Having thus determined r, we have 

q=-S{a + p)y+Fyia-p) 
2q=-{a + p)y-y{a + fi) + y{a-p)-ia-p)y 
= —2ay—2y^. 
Here, of course, we may change the sign of y, and write the solution 
of aq = qb 

in the form q = ay + yfi, 

where y is any vector, and 

a = UFa, /3 = UFb. 


To verify this solution, we' see by ( 1 ) that we require only to 
shew that aq =. qB. 

But their common value is evidently 

— y + ay/3. 

It will be excellent practice for the student to represent the terms 
of this equation by versor-arcsj as in § 54, and to deduce the above 
solution from the diagram directly. He will find that the solution 
may thus be obtained almost intuitively. 

184.J No general method of solving quaternion equations of the 
second or higher degrees has yet been found ; in fact, as will be 
shewn immediatelyj even those of the second degree involve (in 
their most general form) algebraic equations of the sixteenth degree. 
Hence, in the few remaining sections of this Chapter we shall con- 
fine ourselves to one or two of the simple forms for the treatment 
of which a definite process has been devised. But firsts let us 
consider how many roots an equation of the second degree in an 
unknown quaternion must generally have. 

If we substitute for the quaternion the expression 
w-\rix-vjy + hz (§80), 
and treat the quaternion constants in the same way, we shall have 
(§ 80) four equations, generally of the second degree, to determine 
w, X, y, z. The number of roots will therefore be 2* or 16. And 
similar reasoning shews us that a quaternion equation of the mth 
degree has w* roots. It is easy to see, however, from some of the 
simple examples given above (§§ 175-178, &c.) that, unless the 
given equation is equivalent to four scalar equations, the roots will 
contain one or more indeterminate quantities. 

185.] Hamilton has effected in a simple way the solution of the 
quadratic ^^ = qa-{- h, 

or the foUowingj which is virtually the same (as we see by taking 
the conjugate of each side), 

(f = aq + h. 

He puts q—\{a + w + p), 

where w is a scalar, and p a vector. 

Substituting this value in the first equation, we get 

a^ ^{iv + pf + 2wa + ap + pa = 2 {a^ -irWa-\- pa) + ^h, 
or (M; + /3)^ + i2p— pa = a^ + 4^. 

If we put Fa = a, S (a^ + 4b) = e, V{a^ + 45) = 2 y, this becomes 
{w + pY + 2Vap = c+2y; 
H a 

100 QUATERNIONS. [l86. 

which, by equating separately the scalar and vector parts, may be 
broken up into the two equations 

26)2 + p2 =: e, 

V[w-\-ci)p = y. 
The latter of these can be solved for p by the process of § 156, or 
more simply by operating at once by S.a which gives the value of 
S{w + a)p. If we substitute the resulting value of p in the former 
we obtain, as the reader may easily prove, the equation 

{w^-a^) (w*_cK>2 + y2)_(ray)2 = 0. 
The solution of this scalar cubic gives six values of w, for each of 
which we find a value of p, and thence a value of q. 

Hamilton shews {Lectures, p. 633) that only two of these values 
are real quaternions, the remaining four being biquaternions, and 
the other ten roots of the given equation being infinite. 

Hamilton farther remarks that the above process leads, as the 
reader may easily see, to the solution of the two simultaneous 
equations q + r = a, 

qr = -b; 
and he connects it also with the evaluation of certain continued 
fractions with quaternion constituents. (See the Miscellaneous Ex- 
amples at the end of the volume.) 

186.] The equation q^ = aq+qb, 

though apparently of the second degree, is easily reduced to the 
first degree by multiplying 6y, and into, q~^, when it becomes 

l=q-^a + bq-'^, 
and may be treated by the process of § 181. 

187.] The equation f' = aqb, 

where a and h are given quaternions, gives 

q{aqb) = {a,qb)q; 
and, by § 54, it is evident that the planes of q and aqh must coincide, 
A little consideration will shew that the solution depends upon 
drawing two arcs which shall intercept given arcs upon each of two 
great circles ; while one of them bisects the other, and is divided by 
it in the proportion oi m: 1. 



1. Solve the following equations: — 

(a.) V.apP = V.ay^. 
{h.) apfip = papj3. 
(c.) ap + pP = y. 
(d.) S.a^p + ^Sap — aVfip = y. 
(e.) p + ap^= afi. 
(/.) ap^p = p^pa. 
Do any of these impose any restriction on the generality of a and j3 ? 

2. Suppose p = ix+Jy + iz, 

and (j)p = aiSip + hjSjp + ckSJcp ; 
put into Cartesian coordinates the following equations : — 
{a.) T4>p=l. 
(b.) Spil>^p=-l. 
(c.) S.p{<t>^-p^)-^p = -l. 
{d.) Tp = T4Up. 

3. If X, p,, V be any three non-coplanar vectors, and 

q = F/xi'.(/)\+ FvX.(j)ix+ V\p,.(\>v, 
shew that q is necessarily divisible by S.\p,v. 
Also shew that the quotient is 

^2-2 6, 

where Vep is the non-commutative part of <^p. 

Hamilton, Elements, p. 442. 

4. Solve the simultaneous equations : — 

Sap =0,1 
^■> S.ap<bp = O.S 


Sap =0,7 
^"•^ Sp<l>p =0.5 

Sap =0,1 
^ ■' S.aipxp = 0. ) 

5. If # = S/3/Sap+ Frp, 

where r is a given quaternion, shew that 
I = S {8.ay,a^asS.fisfi^^^) + ^S{r Fa^a^ . r/S^^i) + SrlS.apr - 2 (/Sar/S/Sr) + SrTr^ 
and m4>-^<T='2{ra^a^S.^^0^<T) + l.r.ar{r^<T.r)+ VarSr- VrStrr. 

Lectures, p. 561. 


6. If [jog'] denote J>q~V'' 

{pqr) „ . S.plqr], 

to''] » {pqr) + lrq']Sp + lpr]Sq + \jip]Sr, 

and (i'?''*) !! '^'F L?***] > 

shew that the following relations exist among any five quaternions 

=jo{qrst) + q{rsip) + r{stpq) + s{tpqr) + t{pqrs), 
and q{prs() =-[rsf\Spq—[stp~\Srq + [tpr'\Ssq—[prs]Stq. 

Elements, p. 492. 

7. Shew that if t^, i|f be any linear and vector functions, and 
a, /3, y rectangular unit-vectors, the vector 

e = V{4>a\lfa + ^fif^ + (j>r^y) 
is an invariant. [This will be immediately seen if we write it in 
the form 6 = F.<^V^p, 

which is independent of the directions of a, )3, y. But it is good 
practice to dispense with V.] 

If # = S^i^Cft 

and y^rp = ^rjiSC-yp, 

shew that this invariant may be expressed as 
-Sr#C or 2F7ji(^fi. 
Shew also that cfi-ijfp—yjfcpp = F6p. 

The scalar of the same quaternion is also an invariant, and may be 
written as —'S2^Sr]r]j^SC(i 

8. Shew that if <^p = aSap + ^Sfip + ySyp, 
where a, ^, y are any three vectors, then 

-<t>-^pS^.afiy = aiSa,p + ^,S^^ + yj,SY^p, 
where a^ = Vfiy, &c. 

9. Shew that any self-conjugate linear and vector function may 
in general be expressed in terms of two given ones, the expression 
involving terms of the second order. 

Shew also that we may write 

(^ +2; = a (OT-|-a;)2 + 5 (ct + a;) (oj +y) + c(<B +^)2, 
where a, I, c, x, y, z are scalars, and ct and co the two given func- 
tions. What character of generality is necessary in tn- and w ? How 
is the solution affected by non-self-conjugation in one or both ? 
10. Solve the equations : — 

(a) q^ = Zqi+lOj. 
{b.) q^ =2q + i. 
(c.) qaq = bq + c. 
(d.) aq = qr = rb. 


11. Shew that ^FVcjyp = mVV(j)-'^p. 

12. If (^ be self-conjugate, and a, /3, y a rectangular system, 

S.Fa(f>ari3(})^Fy(f>y = 0. 

13. (f)\l/ and yj/cp give the same values of the invariants m, m^,m,^. 

14. If <^' be conjugate to <^, <^<^' is self-conjugate. 

1 5. Shew that ( Va&f + ( Y^fff + ( Yye)^ = 26^ 
if a, /3, y be rectangular unit-vectors. 

16. Prove that V^ {<j)—ff)p = —pV^g+2,Vg. 

17. Solve the equations : — 

'{a.) <^2 _ ^ . 

{b.) ^ + x = i^, I 

where one, or two, unknown linear and vector functions are given 
in terms of known ones. (Tait, Proc. JR. S. JE- 1870-71.) 

18. If <^ be a self-conjugate linear and vector function, £ and rj 
two vectors, the two following equations are consequences one of 
the other, viz. : — f _ F.Tj^rj 

V ^ rm 

Si.rj<pri4>^ri /S^.f^^^^^' 
From either of them we obtain the equation 

This, taken along with one of the others, gives a singular theorem 
when translated into ordinary algebra. What property does it give 
of the surface S.p(fip(j)^p = 1 ? 



188.] Having, in the five preceding Chapters, given a brief 
exposition of the theory and properties of quaternions, we intend 
to devote the rest of the work to examples of their practical appli- 
cation, commencing, of course, with the simplest curve and surface, 
the straight line and the plane. In this and the remaining Chapters 
of the work a few of the earlier examples will be wrought out in 
their fullest detail, with a reference to the first five whenever a 
transformation occurs ; butj as each Chapter proceeds, superfluous 
steps will be gradually omitted, until in the later examples the full 
value of the quaternion processes is exhibited. 

189.] Before proceeding to the proper business of the Chapter we 
make a digression in order to give a few instances of applications 
to ordinary plane geometry. These th-e student may multiply in- 
definitely with great ease. 

(a.) Euclid, I. 5. Let a and ^ be the vector sides of an iso- 
sceles triangle ; /3— a is the base, and 
Ta = T/3. 
The proposition will evidently be proved if we shew that 
a(a-^)-i=X/3(/3-a)-i (§ 52). 
This gives a(a-^)-i= (/3— a)-i/3, 

or (^— a)a = /3(a— j3), 

or _a2 = -/32. 

(5.) Euclid, I. 32. Let ABC be the triangle, and let 

u-= = r» 



where y is a unit-vector perpendicular to the plane of the triangle. 
If ^ = 1, the angle GAB is a right angle (§ 74). Hence 

4=^^(§74). Let^ = m^, C=n'l- We have 

UCB = y''UCA, 
Hence UBA=y'^.y''.y^UAB, 

or 1 = y+™+»>. 

That is l-\-m + n =2, 

or A + B+C=ii. 

This is, properly speaking, Legendre's proof ; and might have been 
given in a far shorter form than that above. In fact we have for 
any three vectors whatever, 

/3y a 
which contains Euclid's proposition as a mere particular case. 

(c.) Euclid, I. 35. Let y3 be the common vector-base of the 
parallelograms, a the conterminous vector-side of any 
one of them. For any other the vector-side is a + a?/3 
(§ 28), and the proposition appears as 

Tn{a + xp) = TV^a (§§ 96, 98), 
which is obviously true. 

{d.) In the base of a triangle find the point from which lines, 
drawn parallel to the sides and limited by them, are 
If a, j3 be the sides, any point in the base has the vector 

p = (1— ar)a+a;/3. 
For the required point 

which determines x. 

Hence the point lies on the line 

which bisects the vertical angle of the triangle. 

This is not the only solution, for we should have written 
T(l-a))Ta = Ti»!Tp, 
instead of the less general form above wMck tacitly assumes that 1—x 
and cc are positive. We leave this to the student. 


(e.) If perpendiculars be erected outwards at the middle points 
of tlie sides of a triangle^ each being proportional to 
the corresponding sidcj the mean point of the triangle 
formed by their extremities coincides with that of the 
original triangle. Find the ratio of each perpendicular 
to half the corresponding side of the old triangle that 
the new triangle may be equilateral. 

Let 2a, 2 /3j and 2 (a + y3) be the vector-sides of the triangle, i a 
unit-vector perpendicular to its plane, e the ratio in question. The 
vectors of the corners of the new triangle are (taking the corner 
opposite to 2/3 as origin) 

/Oj = a + eia, 

P2 = 2a + /3 + ei/3, 

P3 = a + /3— ei((a + /3). 
From these 

*(ft + P2 + /'3) = H4a+2;8) = k (2a-|-2(o + /3)), 
which proves the first part of the proposition. 
For the second part, we must have 

^fe— Pi) = ^(P3— Pa) = 2'(pi— Pa). 
Substituting, expanding, and erasing terms common to all, the 
student wUl easily find 3 gS _ j _ 

Hence, if equilateral triangles be described on the sides of any tri- 
angle, their mean points form an equilateral triangle. 

190.] Such applications of quaternions as those just made are of 
course legitimate, but they are not always profitable. In fact, when 
applied to plane problems, quaternions often degenerate into mere 
scalars, and become (§33) Cartesian coordinates of some kind, so 
that nothing is gained (though nothing is lost) by their use. Before 
leaving this class of questions we take, as an additional example, the 
investigation of some properties of the ellipse. 

191.] We have already seen (§31 {h)) that the equation 
p = acos5 + /3sinfl 

represents an ellipse, Q being a scalar which may have any value. 
Hence, for the vector-tangent at the extremity of p we have 

Ap • ^ ^ „ 

OT = -^ = — asmd + i3cos0, 

which is easily seen to be the value of p when 6 is increased by - • 

Thus it appears that any two values of p, for which difiers by 



- , are conjugate diameters. The area of the parallelogram circum- 
scribed to the ellipse and touching it at the extremities of these 
diameters is, therefore, by § 96, 

^TFp-^ = 4yr(acos0 + /3sin9)(— asine + /3eose) 

= 4yFa/3, 

a constant, as is well known. 

193.] For equal conjugate diameters we must have 

y(aeose + /3sin0) = y(— a sin 9 4-/3 cos 0), 

or (a^— /3^) (cos^^— sin20) + 4^a/3cosesini9 = 0, 

a^ — B^ 
or tan 2 9 = „ „ '^ • 

The square of the common length of these diameters is of course 

a2 + ^^ 


because we see at once from § 191 that the sum of the squares of 
conjugate diameters is constant. 

193.] The maximum or minimum of p is thus found ; 

dTp 1_ dp_ 

de ~~T^''dd' 

= — -^{ — (a^— 13^) cose sine + Sap icos^e—sm^0)). 

For a maximum or minimum this must vanish *, hence 

tan 2^= -5 — ^„, 
a^ — ^^ 

and therefore the longest and shortest diameters are equally inclined 

to each of the equal conjugate diameters. Hence, also, they are at 

right angles to each other. 

194.] Suppose for a moment a and ;3 to be the greatest and least 

semidiameters. Then the equations of any two tangent-lines are 
p = a cos ^ + ;8 sin 5 + «(— a sin ^ + /3 cos 6), 
p = a cos ^1 H- j3 sin 0^ + Xy(^—a sin ^j -)- /3 cos 0-^. 

If these tangent-lines be at right angles to each other 
<?(— asin(9-|-/3cosfl)(— asin^i + /3cosei) = 0, 
or o? sin 6 sin 6^ -)- /3^ cos 5 cos ^j = 0. 

A little reflection will shew him that the latter equation involves an absurdity. 

The student must carefully notice that here we put -j— = 0, and not ^ = 0. 


108 QUATEEJSriOKS. [195. 

Also, for their point of intersection we have, by comparing coeffi- 
cients of a, /3 in the above values of p, 

cos 6—xsmd = cos 6^ —x^ sin 6^ , 

sin O + x cos 6 = sin 6^ + x^ cos d-y . 
Determining x-y from these equations, we easily find 

the equation of a circle ; if we take account of the above relation 
between 6 and d^. 

Also, as the equations above give x = — x^, the tangents are equal 
multiples of the diameters parallel to them ; so that the line joining 
the points of contact is parallel to that joining the extremities of 
these diameters. 

195.] Finally, when the tangents 

p = acosd +y3 sin5 +x (— asinfl +;Scos0), 
p = a cos 0^ + j8 sin d^ + x^ (—a sin 6^ + ^ cos 0^), 
meet in a given point p = aa + bp, 
we have a = cos 6 — x sin = cos 6^ — x^ sin 0^, 

h = sin 0-\-x cos = sin 0^ + x^ cos 0-^ . 
Hence x"^ = a^ + b^—1 = xl 

and a cos + b sin = 1 = a cos ^j + J sin 0^ 

determine the values of and x for the directions and lengths of 
the two tangents. The equation of the chord of contact is 
p = y{a cos 6 + /3 sin 6) + (1 —y) (a cos ^^ + /3 sin 0^). 
If this pass through the point 

p=jia + q^, 
we have ^ = ycos0 + (l— j^)cos9i, 

q = 2/svD.0 + {\ —y) sin 0^, 
from which, by the equations which determine and 5, , we get 

a])-irl(i=yJr\—y= 1. 
Thus if either a and h, or ^ and ^, be given, a linear relation con- 
nects the others. This, by § 30, gives all the ordinary properties of 
poles and polars. 

196.] Although, in §§ 28-30, we have already g^ven some of the 
equations of the line and plane, these were adduced merely for their 
applications to anharmonic coordinates and transversals ; and not 
for investigations of a higher order. Now that we are prepared to 
determine the lengths and inclinations of lines we may investigate 
these and other similar forms anew. 


197.] The equation of the indefinite line d/rawn through the origin 
0, of which the vector OA, = a, forms apart, is evidently 

p = soa, 
or p II a, 

or Fap = 0, 

or Up =: Ua; 

the essential characteristic of these equations being that they are 
linear, and involve one indeterminate scalar in the value of p. 

We may put this perhaps more clearly if we take any two 
vectors, /3, y, which, along with a, form a non-coplanar system. 
Operating with S.Va^ and S.Vay upon any of the preceding equa- 
tions, we get S.afip = 0,1 

and S.ayp = Q.\ *■ '' 

Separately, these are the equations of the planes containing a, /3, 
and o, y ; together, of course, they denote the line of intersection. 

198.] Conversely, to solve equations (1), or to find p in terms of 
known quantities, we see that they may be written 

-S.pFa/3 = 0,-) 
S.pFay = 0,) 
so that p is perpendicular to Fa^ and Fay, and is therefore parallel 
to the vector of their product. That is, 

pII KFajSray, 
II -aS.a^y, 
or p = xa. 
199.] By putting p— ;3 for p we change the origin to a point S 
where 0£ = — ^, or ^0 = y3 ; so that the equation of a line parallel 
to a, and passing through the extremity of a vector /3 drawn from 
the origin, is p—^ = xa, 

or p = p + xa. 

Of course any two parallel lines may be represented as 
p = /3 +xa, 
p = pj^+Xj^a; 
or Fa{p-fi) = 0, 
Fa{p-I3,) = 0. 
200.] The equation of a line, drawn through the extremity of ^, and 
meeting a perpendicularly, is thus found. Suppose it to be parallel 
to y, its equation is p = ^ + xy. 

To determine y we know, first, that it is perpendicular to o, which 
gives Say = 0- 


Secondly, o, ^, and y are in one plane, which gives 

S.a^y - 0. 
These two equations give y |{ r.aFaj3, 
whence we have p =: j3 + soa Vafi. 

This might have been obtained in many other ways ; for instance, 
we see at once that 

/3 = a- la/3 = a-^Safi + a-Wa^. 
This shews that ar^Va^ (which is evidently perpendicular to a) 
is coplanar with a and /3, and is therefore the direction of the re- 
quired line ; so that its equation is 

p = fi+ya-WaP, 

the same as before if we put — ^-5- for x. 


201.J By means of the last investigation we see that 

is the vector perpendicular drawn from the extremity of /3 to the 
line p = xa. 

Changing the origin, we see that 

is the vector perpendicular from the extremity of /3 upon the line 

p = y + xa. 
203.] The vector joining £ (where OJS = fi) with any point in 
p =■ y + xa 
is y + Xa—p. 

Its length is least when 

dT{y+xa—0) = 0, 
or Sa{y + xa—^) = 0, 

i. e. when it is perpendicular to a. 
The last equation gives 

■xa^+Sa{y-^) = 0, 
or xa=—a'' ^Sa{y — /3) . 

Hence the vector perpendicular is 

or a-^Fa{y-fi)=-a-^Fa{l3—y), 

which agrees with the result of last section, 

203.] To find the shortest vector distance between two lines 
p = fi + xa, 
and Pi=/?i + «'iai; 


we must put dT{p—p^ =*0, 

or S{p-p^){dp-dp{) = 0, 
or S{p—pj){adx—aidxi) = 0. 
Since x and x^ are independent^ this breaks up into the two eon- 
'litioiis Sa{p-p,)=0, 

Sajip-pj) = 0; 
proving the well-known truth that the required line is perpendicular 
to each of the given lines. 

Hence it is parallel to Faa-^, and therefore we have 

p—pi—l3 + xa—l3-^—x^ai = yFaai (1) 

Operate by S.aaj and we get 

This determines y, and the shortest distance required is 

[_N'ote. In the two last expressions T before S is inserted simply 
to ensure that the length be positive. If 

/S'.aai(/3 — /3j) be negative, 
then (§89) xS'.a^a(/3— ySj) is positive. 

If we omit the T, we must use in the text that one of these two ex- 
pressions which is positive.J 

To find the extremities of this shortest distance, we must operate 
on (1) with S.a and S.a^. We thus obtain two equations, which 
determine x and x^, as y is already known. 

A somewhat different mode of treating this problem will be dis- 
cussed presently. 

204.] In a given- tetrahedron to find a set of rectangular coordinate 
axes, such that each axis shall ^ass through a pair of opposite edges. 

Let a, /3, y be three (vector) edges of the tetrahedron, one corner 
being the origin. Let p be the vector of the origin of the sought 
rectangular system, which may be called i, j, k (unknown vectors). 
The condition that i, drawn from p, intersects a is 

S.iap = (1) 

That it intersects the opposite edge, whose equation is 

7^ = ^ + x{^-y), 
the condition is 

S.i(fi-y){p-^)=0, or Si{{^-y)p-M = (^ (2) 

There are two other equations hke (1), and two like (2), which can 
be at once written down. 

^^^ QUATERNIONS. [205. 

Put p-y=a^, y-a = ^i, a-fi = y^, 

r^y = a^, Fya = /a^, Ta/S = y^, 

and the six become 

S.iap = 0, S.ia^p —Sia^ = 0, 

S.Jpp = 0, S.j0,p-8jp^ = 0, 

S.kyp = 0, S.hy-^p-Shy^ = 0. 

The two in i give i \\ aSa^-piSaa^ + Sarfi). 

J\\^Sfi2p-p{Sfi^2 + SM> and i\\YSy2P-p{Syy^ + Sysp). 
The conditions of rectangulaffity, viz., 

SiJ = 0, SJi = 0, SM = 0, 
at once give three equations of the fourth order, the first of which is 
= Safi Sa^p S^^p-Sap Sa^{Sfifi^ + Sj3^p)-Sfip Sp^p{Saa^ + Sa^p) 

+ p2 [Saa^ + Sa^p) {S^p^ + Sj3^). 

The required origin of the rectangular system is thus given as 
the intersection of three surfaces of the fourth order. 

205.] The equation Sap = 

imposes on p the sole condition of being perpendicular to a ; and 
therefore, being satisfied by the vector drawn from the origin to 
any point in a plane through the origin and perpendicular to a, is 
the equation of that plane. 

To find this equation by a direct process similar to that usually 
employed in coordinate geometry, we may remark that, by § 29, we 
may write p = xj3 +yy, 

where /3 and y are any two vectors perpendicular to a. In this 
form the equation contains two indeterminates, and is often useful ; 
but it is more usual to eliminate them, which may be done at once 
by operating by Sm, when we obtain the equation first written. 

It may also be written, by eliminating one of the indeterminates 
only, as T^p = ya, 

where the form of the equation shews that Sa^ = 0. 

Similarly we see that 

Sa (p-/3) = 
represents a plane drawn through the extremity of ^ and perpen- 
dicular to a. This, of course, may, hke the last, be put into various 
equivalent forms. 

306,] The line of intersection of the two planes 
8.a (p-/3) = 0, ) 
and 5.ai(p-^i)=0,) ^ > 


contains all points whose value of p satisfies both conditions. But 
we may write (§ 92), since a, a^, and Faa-^ are not coplanar, 

pS.aa-^Vaa-^^ — Vaa^SMa^p^ V.a-J^aai8ap+ F.F{aa^)aSa^p, 
or, by the given equations, 

—pT^ Vaa-^ = r.d^ Vaa^Sa^ + V. r{aa^ aSa^^ + x Yaa^, (2) 

where x, a scalar indeterminate, is put for S.aa^p which may have 
any value. In practice, however, the two definite given scalar 
equations are generally more useful than the partially indeterminate 
vector-form which we have derived from them. 

When both planes pass through the origin we have /3 = /S^ = 0, 
and obtain at once ^ ^ ^ jr^^ 

as the equation of the line of intersection. 

207.] The plane passing through the origin, and through the line of 
intersection of the two planes (1), is easily seen to have the equation 
Sa^^iSap — SajSSaip = 0, 
or S{aSa^l3-^—a-^SaP)p = 0. 

For this is evidently the equation of a plane passing through the 
origin. And^ if p he such that 

Sap = Safi, 
we also have Saj^p = Sa^^^, 

which are equations (1). 

Hence we see that the vector 

aSa^^jSi — ajSaj3 
is perpendicular to the vector-line of intersection (2) of the two 
planes (1), and to every vector joining the origin with a point in 
that line. 

The student may verify these statements as an exercise. 

208.] To find the vector-perpendicular from the extremity of ^ on 
the plane Sap = 0, 

we must note that it is necessarily parallel to a, and hence that the 
value of p for its foot is p — ^^^a, 
where xa is the vector-perpendicular in question. 

Hence Sa {j3 + xa) = 0, 

which gives xa^:= —Sa^, 

or Xa = —a~^Sa/3. 

Similarly the vector-perpendicular from the extremity of /3 on the 

may easily be shewn to be 



114 QUATERNIONS. [209. 

209.] The equation of the plane which passes through the ecctremities 
of a,^,y may be thus found. If p be the vector of any point in it) 
p—a, a—p, and /3— y lie in the plane, and therefore (§101) 
or Sp{ra^-{- Vfiy+ rya)-S.a^y = 0. 

Hence, if 6 = a; ( F"a/3 + T/Sy + Fya) 

be the vector-perpendicular from the origin on the plane containing 
the extremities of a, y3, y, we have 

6 = (ra/3+ r/3y+ Fyay^S.a^y. 
From this formula, whose interpretation is easy, many curious pro- 
perties of a tetrahedron may be deduced by the reader. Thus, for 
instance, if we take the tensor of each side, and remember the 
result of § 100, we see that 

is twice the area of the base of the tetrahedron. This may he more 
simply proved thus. The vector area of base is 

ir{d-fi) (y-/3) =-iiral3+ r^y+ Fya). 
Hence the sum of the vector areas of the faces of a tetrahedron, 
and therefore of any solid whatever, is zero. This is the hydrostatic 
proposition for solids immersed in a fluid subject to no external 

310.] Taking any two lines whose equations are 
p = 13 + xa, 

p =: jSj^ + X^Oi, 

we see that S.aaj(p — 6) ^ 

is the equation of a plane parallel to both. Which plane, of course, 

depends on the value of 8. 

Now if 8 = /3, the plane contains the first line ; if 8 = ^1, the 

Hence, liyVaa^ be the shortest vector distance between the lines, 
we have 5_„„^ {fi-^^-y Faa^) = 0, 

or TiyFaa^) = m(/3-^i) UFaa^, 

the result of § 203. 

211.J Find the equation of the plane, passing through the origin, 
which makes equal angles with three given lines. Also find the angles 
in question. 

Let a, y3, y be unit-vectors in the directions of the lines, and let 
the equation of the plane be 

Sbp = 0. 


Then we have evidently 

Sab = /S/38 = Syb = x, suppose, 

where ^ 


is the sine of each of the required angles. 

But (§ 92) we have 

bS.a/Sy = X iFa^+ F^y+ Fya). 
Hence S.p ( VajS + V/3y + Fya) = 

is the required equation ; and the required sine is 


~ T{ra^+rpy+rya)' 
312.] Find the locus of the middle points of a series of straight 
lines, each parallel to a given plane and having its extremities*in two 
fixed lines. 

Let 8yp — 

be the plane, and p = yg^a-^^ ^^ ^-y^x^a^, 

the fixed lines. Also let x and x-^ correspond to the extremities of 
one of the variable lines, is- being the vector of its middle point. 
Then, obviously, 2-a! = ^ + xa-\-^^+x-^a^. 
Also 8y{fi—^^->rXa—x^a^ = 0. 

This gives a linear relation between so and x-^ , so that, if we sub- 
stitute for Xj^ in the preceding equation, we obtain a result of the 
form ^^8+a;e, 

where 8 and e are known vectors. The required locus is, therefore, 
a straight line. 

313.] Three planes meet in a point, and through the line of inter- 
section of each pai/r a plane is drawn perpendicular to the third ; prove 
that, in general, these planes pass through the same line. 

Let the point be taken as origin, and let the equations of the planes 

^e Sap = 0, Sl3p = 0, Syp = 0. 

The line of intersection of the first two is || FajB, and therefore the 

normal to the first of the new planes is 

Hence the equation of this plane is 

S.pF.yFa^ = 0, 
or SfipSay—SapS^y = 0, 

and those of the other two planes may be easily formed from this 
by cyclical permutation of a, /3, y. 

I a 

116 QUATERNIONS. [214. 

We see at once that any two of these equations give the third by 
addition or subtraction, which is the proof of the theorem. 

214.] Griven any number of points A, B, G, 8fc., whose vectors 
{from the origin) are a^, Og, a.^, 8fc.,find the plane through the origin 
for which the sum of the squares of the perpendiculars let fall upon it 
from these points is a maximum or minimum. 

Let ^wp = 

be the required equation, with the condition (evidently allowable) 

IW= 1. 
The perpendiculars are (§ 208) — ■nr~^*S'OTai, &c. 

Hence ^S^-^a 

is a maximum. This gives 

"^.SisaSadiTt! = ; 
and the condition that ■zn- is a unit-vector gives 

SnydvT = 0. 

Hence, as d-sr may have any of an infinite number of values, these 
equations cannot be consistent unless 

where a; is a scalar. 

The values of o are known, so that if we put 

^ is a given self-conjugate linear and vector function, and therefore 
a; has three values {g^, g^, g^, § 164) which correspond to three 
mutually perpendicular values of -57. For one of these there is a 
maximum, for another a minimum, for the third a maximum- 
minimum, in the most general case when g^, g.^., g^ are all different. 

215.] The following beautiful problem is due to MaccuUagh. 
Of a system of three rectangular vectors, passing through the origin, 
two lie on given planes, find the locus of the third. 

Let the rectangular vectors be ot, p, a. Then by the conditions 
of the problem gsyp = Spa = Sa^ = 0, 

and iSara- = 0, S^p = 0. 

The solution depends on the elimination of p and ot among these 
five equations. [This would, in general, be impossible, as p and ■in- 
between them involve six unknown scalars ; but, as the tensors are 
(by the very form of the equations) not involved, the five given 
equations are necessary and suflicient to eliminate the four unknown 
scalars which are really involved. Formally to complete the re- 
quisite number of equations we might write 
Ts^ = a, Tp = h, 
but a and h may have any values whatever.] 


From Sasr = 0, /So-sr = 0, 

we have in- = xFaa: 

Similarly, from Sfip = 0, Sap = 0, 

we have P = y Vj3a: 

Substitute in the remaining equation 

S'srp = 0, 
and we have S.FaaF^a = 0, 

or Sa<rSj3<T — cr^Sa^ = 0, 

the required equation. As will be seen in next Chapter, this is a 
cone of the second order whose circular sections are perpendicular 
to a and /3. [The disappearance of x and y in the elimination in- 
structively illustrates the note above.J 


1. What propositions of Euclid are proved by the mere /by»« of 
the equation p = {l—ai)a + xj3, 

which denotes the line joining any two points in space ? 

2. Shew that the chord of contact, of tangents to a parabola 
which meet at right angles, passes through a fixed point. 

3. Prove the chief properties of the circle (as in Euclid, III) from 
the equation p = a cos + ^ sin ; 

where Ta = Tfi, and Sa^ = 0. 

4. What, locus is represented by the equation 

S^ap + p^= 0, 
where Ta= I? 

5. What is the condition that the lines 

Fap = A Fa^P = ySi, 

intersect? If this is not satisfied, what is the shortest distance 
between them ? 

6. Find the equation of the plane which contains the two parallel 
lines ra(p-/3)=0, Fa{p-^i) = 0. 

7. Find the equation of the plane which contains 

ra(p-/3) = 0, 
and is perpendicular to gyp — o. 

8. Find the equation of a straight line passing through a given 
point, and making a given angle with a given plane. 

Hence form the general equation of a right cone. 


9. What conditions must be satisfied with regard to a number of 
given lines in space that it may be possible to draw through each 
of them a plane in such a way that these planes may intersect in a 
common line ? 

10. Find the equation of the locus of a point the sum of the 
squares of whose distances from a number of given planes is con- 

11. Substitu^ "lines" for "planes" in (10). 

12. Find the equation of the plane which bisects, at right angles, 
the shortest distance between two given lines. 

Find the locus of a point in this plane which is equidistant from 
the given lines. 

1 3. Find the conditions that the simultaneous equations 

Sap = a, S^p = 6, Syp = c, 

may represent a line, and not a point. 

1 4. What is represented by the equations 

{Sapf = {Sl3py^ = {Syp)^ 
where a, /3, y are any three vectors ? 

15. Find the equation of the plane which passes through two 
given points and makes a given angle with a given plane. 

16. Find the area of the triangle whose corners have the vectors 
a, /3, y. 

Hence form the equation of a circular cylinder whose axis and 
radius are given. 

17. (Hamilton, Bishop Law's Fremium Ex., 1858). 

{a.) Assign some of the transformations of the expression 

/3— a' 

where a and /3 are the vectors of two given points A and B. 
{h.) The expression represents the vector y, or OC, of a point C 

in the straight line AB. 
(c.) Assign the position of this point C. 

18. (Ibid.) 

(a.) If a, /3, y, 8 be the vectors of four points. A, B, C, B, what 
is the condition for those points being in one plane ? 

(h.) When these four vectors from one origin do not thus ter- 
minate upon one plane, what is the expression for the 
volume of the pyramid, of which the four points are the 
corners ? 

(c). Express the perpendicular S let fall from the origin on 
the plane ABC, in terms of a, y3, y. 


19. Find the locus of a point equidistant from the three planes 

Sap = 0, S^p = 0, Syp = 0. 

20. If three mutually perpendicular vectors be drawn from a 
point to a plane, the sum of the reciprocals of the squares of their 
lengths is independent of their directions. 

21. Find the general form of the equation of a plane from the 
condition (which is to be assumed as a definition) that any two 
planes intersect in a single straight line. 

22. Prove that the sum of the vector areas of the faces of any 
polyhedron is zero. 



216.] Aftee that of the plane the equations next in order of 
simplicity are those of the sphere, and of the cone of the second 
order. To these we devote a short Chapter as a valuable prepara- 
tion for the study of surfaces of the second order in general. 

217.] The equation y^ _ ^a 

or p^ = (^, 
denotes that the length of p is the same as that of a given vector a, 
and therefore belongs to a sphere of radius Ta whose centre is the 
origin. In § 107 several transformations of this equation were ob- 
tained, some of which we will repeat here with their interpretations. 
Thus ^(p + a)(p-a) = 

shews that the chords drawn from any point on the sphere to the 
extremities of a diameter (whose vectors are a and —a) are at right 
angles to each other. 

r(p + a)(p-a)= iTVap 
shews that the rectangle under these chords is four times the area 
of the triangle two of whose sides are a and p. 

(0 = (p + a)"^a(/3 + a) (see § 105) 
shews that the angle at the centre in any circle is double that at 
the circumference standing on the same arc. All these are easy 
consequences of the processes already explained for the interpretation 
of quaternion expressions. 

218.] If the centre of the sphere be at the extremity of a the 
equation may be written 

T{p-a) = Tp, 
which is the most general form. 

If Ta = T/3, 

or a2 = /3^ 


in which ease the origin is a point on the surfaee of .the sphere, this 
becomes p^-2Sap = 0. 

From this, in the form 

Sp{p — 2a) = 
another proof that the angle in a semicircle is a right angle is de- 
rived at once. 

219.] The converse problem is — Mnd the locus of tJiefeet of per- 
pendiculars let fall from a given point (p=/3) on planes passing through 
the origin. 

Let Sap = 

be one of the planes, then (§208) the vector-perpendicular is 

— a-^Saj3, 
and, for the locus of its foot, 

p = /3 — a-i/S'a/3, 
= orWap. 
[This is an example of a peculiar form in which quaternions some- 
times give us the equation of a surfaee. The equation is a vector 
one, or equivalent to three scalar equations ; but it involves the 
undetermined' vector a in such a way as to be equivalent to only- 
two indeterminates (as the tensor of a is evidently not involved). 
To put the equation in a more immediately interpretable form, a 
must be eliminated, and the remarks just made shew this to be 

Now {p-^Y =a-Wap, 

and (operating by S.fi) 

Adding these equations, we get 

P^-S^P = 0, 

so that, as is evident, the locus is the sphere of which y3 is a dia- 

220.] To find the intersection of the two spheres 
T(p-a) = h, 
and ^(p-«i) = ^/3i. 

square the equations, and subtract, and we have 

2S{a-ai)p = a^-ai^-{^^-l3j^), 
which is the equation of a plane, perpendicular to a— aj the vector 
joining the centres of the spheres. This is always a real plane 
whether the spheres intersect or not. It is, in fact, what is called 
their Radical Plane. 

122 QUATERNIONS. [221. 

331.] Find the locus of a point the ratio of whose distances from 
two given points is constant. 

Let the given points be and A, the extremities of the vector a. 
Also let P be the required point in any of its positions, and OP=p. 

Then, at once, if n be the ratio of the lengths of the two lines, 
T{p-a) = nTp. 
This gives p^ — 2Sap + a^ = »2 p^, 

or, by an easy transformation, 

Thus the locus is a sphere whose radius is Tf- ^^> and whose 

centre is at JB, where 0£ = 5- > a definite point in the line OA. 

1—n^ ^ 

632.] ^in any line, OP, drawn from the origin to a given plane, 

OQ be taken such that OQ.OP is constant, fnd the locus of Q. 

Let Sap = a 

be the equation of the plane, ct a vector of the required surface. 

Then, by the conditions, 

T'HT Tp = constant = 5^ (suppose), 

and Z7«r = Up. 

From these p = -s= — = 5- • 

Substituting in the equation of the plane, we have 

aw^ + b^Saw = 0, 
which shews that the locus is a sphere, the origin being situated on 
it at the point farthest from the given plane. 

333.] FiMd the locus of points the sum of the squares of whose dis- 
tances from a set of given points is a constant quantity. Find also the 
least value of this constant, and the corresponding locus. 

Let the vectors from the origin to the given points be oj, Oj, 

a„, and to the sought point p, then 

-c2 = {p-c^f+[p-a^f + + (p-a„)^ 

= np^-2Sp'2a+-S,{a^). 

Otherwise (,_^«/= _ flilli^!! + (?#, 

\ n' n n^ 

the equation of a sphere the vector of whose centre is — > i.e. 
whose centre is the mean of the system of given points. 

Suppose the origin to be placed at the mean point, the equation 

becomes /.2 j. y („i\ 

p2 ^ _^__+±S5l1 (for 2a = 0, § 31 (e)). 


The right-hand side is negative, and therefore the equation denotes 
a real surface, if ^2 ^ 2Ta^ 

as might have been expected. When these quantities are equal, 
the locus becomes a point, viz. the new origin, or the mean point of 
the system. 

334.J If we differentiate the equation 

Tp = Ta 
we get Spdp — 0. 

Hence {^ \i7), p is normal ^ the surface at its extremity, a well- 
known property of the sphere. 

If tn- be any point in the plane which touches the sphere at the 
extremity of p, ta-— p is a line in the tangent plane, and therefore 
perpendicular to p. So that 

8p{'7!-p) = 0, 

or S-arp = — Tp^ = a^ 

is the equation of the tangent plane. 

225 .J If this plane pass through a given point B, whose vector 
is fi, we have ^^^ ^ „2. 

This is the equation of a plane, perpendicular to /3, and cutting 
from it a portion whose length is 

Tp ' 
If this plane pass through a fixed point whose vector is y we must 
have spy = a^ 

so that the locus of /8 is a plane. These results contain all the 
ordinary properties of poles and polars with regard to a sphere. 

226.] A line drawn parallel to y, from the extremity of /3, has 
the equation p — a^^ 

This meets the sphere p2 _ ^2 

in points for which w has the values given by the equation 

P^ + 2xSl3y-^x^y'^ = a^. 
The values of a; are imaginary, that is, there is no intersection, if 

The values are equal, or the line touches the sphere, if 
aV+^^/3y = 0, 
or S^l3y = y^P^-a^). 
This is the equation of a cone similar and similarly situated to the 
cone of tangent-lines drawn to the sphere, but its vertex is at the 
centre. That the equation represents a cone is obvious from the 

124 QUATERNIONS. [227, 

fact that it is homogeneous in Ty, i.e. that it is independent of the 
length of the vector y. 

[It may be remarked that from the form of the above equation 
we see that, if x and x' be its roots, we have 
which is Euclid, III, 35, 36, extended to a sphere.] 

227.] Find the locus of the foot of the perpendicular let fall from 
a given point of a sphere on any tangent-plane. 

Taking the centre as origin, the equation of any tangent-plane 
may be written ^^p ^ „2_ 

The perpendicular must be parallel to p, so that, if we suppose it 
drawn from the extremity of a (which is a point on the sphere) we 
have as one value of ■or 

■cT = a-\-xp. 

From these equations, with the help of that of the sphere 

we must eliminate p and x. 

We have by operating on the vector equation by S.'^ 

■sr^ = SaiiT+xS'STp 

■=■ /iSatsr + ara^. 

__ CT — a a^ (■or — a) 
Hence p = = — 5 — 5 

Taking the tensors, we have 

(i!r2_^a^)2 = a2(ti^-a)^ 
the required equation. It may be put in the form 

and the interpretation of this gives at once a characteristic property 
of the surface formed by the rotation of the Cardioid about its axis 
of symmetry. 

228.] We have seen that a sphere, referred to any point what- 
ever as origin, has the equation 

T{p-a) = T^. 
Hence, to find the rectangle under the segments of a chord drawn 
through any point, we may put 

where y is any unit-vector whatever. This gives 

x^y^-2xSay+a^ = ^^, 
and the product of the two values of x is 



.... • • 

This is positive, or the vector-chords are drawn in the same direc- 
tion, if T&<Ta, 
i.e. if the origin is outside the sphere. 

229.] A, B are fixed points s and, leing the origin and P a point 
m space, jjp2 ^ ^pa = Qpa . 

find the locus ofP, and explain the result when LAOB is a right, or 
an obtuse, angle. 

Let OJ = a, 0B = ^,6P=p, then 

or p2_2^(a + y3)p=_(a2+/32), 
or y{p-(a4-/3)}=^/(-2&/3). 
While Sa^ is negative, that is, while LAOB is acute, the locus is a 

sphere whose centre has the vector o + /3. If ASa/3=0, or LAOB=-, 
the locus is reduced to the point 

p = a + /3. 
"If LAOB>- there is no point which satisfies the conditions. 

230.] Bescriie a sphere, with its centre in a given line, so as to 
pass through a given point and touch a given plane. 

Let xa, where « is an undetermined scalar, be the vector of the 
centre, r the radius of the sphere^ /3 the vector of the given point, 
and Syp = a 

the equation of the given plane. 

The vector-perpendicular from the point xa on the given plane is 
(§208) {a-xSya)y-''. 

Hence, to determine x we have the equation 

T.{a-x8ya)y-'^ = T{xa-^) = r, 
so that there are, in general, two solutions. It will be a good 
exercise for the student to find from this equation the condition 
that there may be no solutioQj or two coincident ones. 

231.] Bescribe a sphere whose centre is in a given line, and which 
passes through two given points. 

Let the vector of the centre be xa, as in last section, and let the 
vectors of the points be ^ and y. Then, at once, 

T{y-xa) =T{fi-xa) = r. 
Here there is but one sphere, except in the particular case when we 
have Ty = T^, and Say = Sa^, 

in which case there is an infinite number. 

126 QUATERNIONS. [232. 


The student should carefiiUy compare the results of this section 
and the last, so as to discover why in general two solutions are 
possible in the one case, and only one in the other. 

232.] A sphere touches each of two straight lines, which do not 
meet -. find, the locus of its centre. 

We may take the origin at the middle point of the shortest dis- 
tance (§203) between the given lines, and their equations will then 
be p = a-\-x^, 

where" we have, of course, 

Sa^ = 0, xSa/3i = 0. 
Let <r be the vector of the centre, p that of any point, of one of the 
spheres, and r its radius ; its equation is 

T{p-a) = r. 
Since the two given lines are tangents, the following equations in x 
and Xi must have pairs of equal roots, 

2'(a4-«/8 — (7) = r, 
T{-a + a;^Pi-a-)=zr. 
The equality of the roots in each gives us the conditions 
S^I3<T =/32((a-(r)2+»-2), 
-S2/3i<T=^f((a + cr)2+r2). 
Eliminating r we obtain 

^-^S^fia-fil^S^fi^a- = (a-o-)2-(a + <r)2 =-45a(7, 
which is the equation of the required locus. 

[As we have not, so far, entered on the consideration of the qua- 
ternion form of the equations of the various surfaces of the second 
order, we may translate this into Cartesian coordinates to find its 
meaning. If we take coordinate axes of so, y, z respectively parallel 
to |3, /3i, a, it becomes at once 

{x-\-myf^{jl-\-mxf' =^ pz, 
where m and p are constants ; and shews that the locus is a hy- 
perbolic paraboloid. Such transformations, which are exceedingly 
simple in all cases, will be of frequent use to the student who is 
proficient in Cartesian geometry, in the early stages of his study of 
quaternions. As he acquires a practical knowledge of the new 
calculus, the need of such assistance will gradually cease to be 

Simple as the above solution is, quaternions enable us to give one 
vastly simpler. For the problem may be thus stated — Find the 
locus of the point whose distances from two given, lines are equal. 


And, with, the above notation, the equality of the perpendiculars is 
expressed (§ 201) by 

TV. (a -a)U^ = TV. (a + <t) U^^ , 
which is easily seen to be equivalent to the equation obtained above. 

233.] Two spheres being given, shew that spheres which cut them at 
given atigles cut at right angles another fixed sphere. 

If be the distance between the centres of two spheres whose radii 
are a and i, the cosine of the angle of intersection is evidently 

Hence, if a, a^, and p be the vectors of the centres, and «,«!,»• the 
radii, of the two fixed, and of one of the variable, spheres ; A and 
^1 the angles of intersection, we have 

{p — af+a'^-\-r^= 2ar cos A, 
{p—aj)^ +al+r^ = 2ajrcosAj^. 
Eliminating the first power of r, we evidently must obtain a result 
sueh as (p— /S)^ + h^ + r^ = 0, 

where (by what precedes) /3 is the vector of the centre, and b the 
radius, of a fixed sphere 

{p-l3)^ + b^ = 0, 

which is cut at right angles by all the varying spheres. By effect- 
ing the elimination exactly we easily find b and y3 in terms of given 

234.J To inscribe in a given sphere a closed polygon, plane or 
gauche, whose sides shall be parallel respectively to each of a series of 
given vectors. 

Let Tp = 1 

be the sphere, a, fi, y , -q, 6 the vectors, n in number, and let 

Pi,P2, p„ , be the vector-radii drawn to the angles of the polygon. 

Then p2~Pi = ^i"' ^^-f ^^■ 

From this, by operating by S.{p2 + Pi), we get 
P2-Pi = = Sap2 + Sapi. 
Also = Vap2— Fapi. 

Adding, we get = apa + -^"Pi = "Pz + Pi «• 
Hence P2=— a~Vi"- 

[This might have been written down»at once from the result of 

Similarly p^ = — /3~V2/3 = ^"^ °-~^ Pi<^^> ^^• 

Thus, finally, since the polygon is closed, 

P»+i = Pi = i-T&'^rj-^ ^-''a-'p^a^ r,B. 


128 QUATERNIONS. [235. 

We may suppose the tensors of a, )3 t;, 6 to be each unity. 

Hence, if ^ ^ „^ ^g^ 

we have ffl-i = fl-i jj-i /3-1 a-\ 

which is a known quaternion ; and thus our condition becomes 

Pi = (-)"«">]«• 
This divides itself into two cases, according as n is an even or an 
odd number. 

If n be even, we have 

api = pya. 
Removing the common part p-^^Sa, we have 

Fp^Va = 0. 
This gives one determinate direction, ± Fa, for ^ ; and shews that 
there are two, and only two, solutions. 
If n be odd, we have ap^ = —p^a, 
which requires that we have 

Sa = 0, 
i. e. a must be a vector. 

Hence Sap^ = 0, 

and therefore pj^ may be drawn to any point in the great circle of 

the unit-sphere whose poles are on the vector a. 

235.] To illustrate these results, let us take first the ease of m= 3. 
Here we must have S.aBy = 

or the three given vectors must" (as is obvious on other grounds) be 
parallel to one plane. Here afiy, which lies in this plane, is (§ 106) 
the vector-tangent at the first corner of each of the inscribed tri- 
angles; and is obviously perpendicular to the vector drawn from 
the centre to that corner. 

Ifn=4, we have p^ y f . ajSyb, 

as might have been at once seen from §106. 

236.] Hamilton has given {Lectures, p. 674) an ingenious and 
simple process by which the above investigation is rendered ap- 
plicable to the more difficult problem in which each side of the 
inscribed polygon is to pass through a given point instead of being 
parallel to a given line. His process depends upon the integration 
of a linear equation in finite differences. By an immediate appli- 
cation of the linear and veetor function of Chapter V, the above 
solutions may be at once extended to any central surface of the 
second order. 

237.] To find the equation of a cone of revolution, whose vertex is 
the origin. 


Suppose a, where jTa = l , to be its axis, and e the cosine of its 
semi-vertical angle ; then, if p be the vector of any point in the 
cone, SaUpz^^e, 

or S^ap = —e^p^. 

238.] Change the origin to the point in the axis whose vector is 
xa, and the equation becomes 

{ — X + SaTjrY ^—e^i^a + 'ury. 
Let the radius of the section of the cone made by 

Saur = 
retain a constant value &, while m changes ; this necessitates 


Vb^ + m^ 
so that when x is infinite, e is unity. In this case the equation 
becomes ^2„^ ^. ^2 ^ j2 _ q^ 

which must therefore be the equation of a circular cylinder of radius 
b, whose axis is the vector a. To verify this we have only to notice 
that if w be the vector of a point of such a cylinder we must (§201) 
have TFaTu- = b, 

which is the same equation as that above. 

239.] To find, generally, the equation of a cone which has a circular 
section : — 

Take the origin as vertex, and let the circular section be the 
intersection of the plane Sap = 1 

with the sphere (passing through the origin) 

p2 = Sl3p. 

These equations may be written thus, 

SaUp= =-, 

-Tp = S^Up. 
Hence, eliminating Tp, we find the following equation which Up 
must satisfy— SaUpSfiUp =-l, 

or p^—SapS^p = 0, 

which is therefore the required equation of the cone. 

As a and /S are similarly involved, the mere form of this equation 
proves the existence of the subcontrary section discovered by Apol- 

240.] The equation just obtained may be written 

S.UaUpS.Ul3Up = --^, 



or, since a and y3 are perpendicular to the cyclic arcs (§ 59*), 

sinj» sinj!)'= constant, 
where j) and j)' are arcs drawn from any point of a spherical conic 
perpendicular to the cyclic arcs. This is a well-known property of 
such curves. 

241 .J If we cut the cyclic cone by any plane passing through 
the origin, as gyp _ q^ 

then Fay and Ffiy are the traces on the cyclic planes, so that 

p = xUVay+yUF^y (§ 29). 
Substitute in the equation of the cone, and we get 

—x^—^^ + Pxy = 0, 
where P is a known scalar. Hence the values of x and _y are the 
same pair of numbers. This is a very elementary proof of the 
proposition in § 59*, that PL = MQ (in the last figure of that 

243.] When x and ^ are equal, the transversal arc becomes a 
tangent to the spherical conic, and is evidently bisected at the 
point of contact. Here we have 

P=2 = 2S.UrayUrfiy+-^^-''^^'^' 


This is the equation of the cone whose sides are perpendiculars 
(through the origin) to the planes which touch the cyclic cone, and 
from this property the same equation may readily be deduced. 

243.] It may be well to observe that the property of the Stereo- 
graphic projection of the sphere, viz. that the projection of a circle 
is a circle, is an immediate' consequence of the above form of the 
equation of a cyclic cone. 

244 J That § 239 gives the most general form of the equation 
of a cone of the second order, when the vertex is taken as origin, 
follows from the early results of next Chapter. For it is shewn 
in § 249 that the equation of a cone of the second order can always 
be put in the form 2 2.Sap8^p + Ap^ = 0. 
This may be written 8p<pP = 0, 

where <p is the self-conjugate linear and vector function 

(^/) = 2F.ap0 + (A + ^Safi)p. 
By § 168 this may be transformed to 

<i>P=pp+ F. Kpp., 
and the general equation of the cone becomes 
{j)-S\p.)p'^ + 2S\pSf^p = 0, 
which is the form obtained in § 239. 



245.] Taking the form Spct>p = 
as the simplest, we fiad by differentiation 
Sdp(f>p + Spd<pp = 0, 
or '2Sdp(j)p = 0. 

Hence (pp is perpendicular to the tangent-plane at the extremity of 
p. The equation of this plane is therefore (■nr being the vector of 
any point in it) Scj^p (t^-p) = 0, 

or, by the equation of the cone, 

aSct(^P = 0. 
246.] T^e equation of the cone of normals to the tangent-planes of 
a given cone can he easily formed from that of the cone itself. For we 
may write it in the form 

S{<i>-^4,p)<pp = o, 

and if we put <pp-=a; a vector of the new cone, the equation becomes 

■Sa4>-^<T = 0. 
Numerous curious properties of these connected cones, and of the 
corresponding spherical conies, follow at once from these equations. 
But we must leave them to the reader. 

247.] As a final example, let vls find the equation of a cyclic cone 
when five of its vector-sides are given — i. e. find the cone of the second 
order whose vertex is the origin, and on whose surface lie the vectors 
a, A y, S, e. 

If we write 

= s.r{rapvbi)r(r^yrep)F{rybFpai (i) 

we have the equation of a cone whose vertex is the origin — ^for the 
equation is not altered by putting sep for p. Also it is the equation 
of a cone of the second degree, since p occurs only twice. Moreover 
the vectors a, ^,y, 6, e are sides of the cone, because if any one of 
them be put for p the equation is satisfied. Thus if we put /3 for p 
the equation becomes 

= s.v{rafirbe)r{rpyn^)r{rybr^a) 

= S.FiFa^ne) { F^aS.FyhF^yFe^- FybS.FfiaFPyFe^}. 
The first term vanishes because 

S.F{Fa^Fbe)Fl3a= 0, 
and the second because 

S.F^aF^yFflS = 0, 
since the three vectors FjSa, FjSy, Fej3, being each at right angles to 
/3, must be in one plane. 

As is remarked by Hamilton, this is a very simple proof of Pascal's 

K 2, 


Theorem — for (1) is the condition that the intersections of the 
planes of a, /3 and 8, e ; /3, y and e, p; y, 8 and p, a ; shall lie in one 
plane ; or, making the statement for any plane section of the cone, 
that the points of intersection of the three pairs of opposite sides, of 
a hexagon inscribed in a curve, may always lie in one straight line, 
the curve must he a conic section. 


1 . On the vector of a point P in the plane 

Sap= 1 
a point Q is taken, such that QO.OP is constant ; find the equation 
of the locus of Q. 

2. "What spheres cut the loci of P and Q in (1) so that both 
lines of intersection lie on a cone whose vertex is ? 

3. A sphere touches a fixed plane, and cuts a fixed sphere. If 
the point of contact with the plane be given, the plane of the inter- 
section of the spheres contains a fixed line. 

Find the locus of the centre of the variable sphere, if the plane of 
its intersection with the fiied sphere passes through a given point. 

4. Find the radii of the spheres which touch, simultaneously, the 
four, given planes 

Sap = 0, Sj3p = 0, Syp = 0, Sbp = 1. 

[What is the volume of the tetrahedron enclosed by these planes ?] 

5. If a moveable line, passing through the origin, make with 
any number of fixed lines angles 6, 6^, 02, &c., such that 

a cos.O + «! cos.^i + = constant, 

where «, «i, are constant scalars, the line describes a right cone. 

6. Determine the conditions that 

Sp(j)p ^ 
may represent a ri^M cone. 

7. What property of a cone (or of a spherical conic) is given 
directly by the following form of its equation, 

S.ipxp ^ ? 

8. What are the conditions that the surfaces represented by 

Sp^p = 0, and S.ipKp = 0, 
may degenerate into pairs of planes ? 


9. Find the locus of the vertices of all right cones which have a 
common ellipse as base. 

10. Two right circular cones have their axes parallel, shew that 
the orthogonal projection of their curve of intersection on the plane 
containing their axes is a parabola. 

11. Two spheres being given in magnitude and position, every 
sphere which intersects them in given angles will touch two other 
fixed spheres and cut a third at right angles. 

12. If a sphere be placed on a tablcj the breadth" of the elliptic 
shadow formed by rays diverging from a fixed point is independent 
of the position of the sphere. 

1 3. Form the equation of the cylinder which has a given circular 
section, and a given axis. Find the direction of the normal to the 
subcontrary section. 

14. Given the base of a spherical triangle, and the product of 
the cosines of the sides, the locus of the vertex is a spherical conic, 
the poles of whose cyclic arcs are the extremities of the given 

15. (Hamilton, Bishop Law's 'Premium Ex., 1858.) 

(a.) What property of a sphero-conic is most immediately in- 
dicated by the equation 

a p 

{b.) The equation {VKpf + {StipY = 

also represents a cone of the second order ; A. is a focal 
line, and jj. is perpendicular to the director-plane cor- 

(c.) What property of a sphero-conic does the equation most 
immediately indicate ? 

16. Shew that the areas of all triangles, bounded by a tangent 
to a spherical conic and the cyclic arcs, are equal. 

17. Shew that the locus of a point, the sum of whose arcual dis- 
tances from two given points on a sphere is constant, is a spherical 

18. If two tangent planes be drawn to a cyclic cone, the four 
lines in which they intersect the cyclic planes are sides of a right 

19. Find the equation of the cone whose sides are the intersections 
of pairs of mutually perpendicular tangent planes to a given cyclic 


20. Find the condition that five given points may lie on a 

21. What is the surface denoted by the equation 

where p = xa+y^ + zy, 

a, )3, y being given vectors, and x, y, z variable scalars ? 

Express the equation of the surface in terms of p, a, /3, y alone. 

22. Find the equation of the cone whose sides bisect the angles 
between a fixed line and any line, in a given, plane, which meets the 
fixed line. 

What property of a spherical conic is most directly given by 
this result ? 



248.] The general scalar equation of the second order in a vector 
p must evidently contain a term independent of p, terms of the form 
S.apb involving p to the first degree, and others of the form S.aphpc 
involving p to the second degree^ a, h, c, &e. being constant quater- 
nions. Now the term S.apd may be written as 

or as S.{Sa+ ra)p{Sb+ Vb) = SaSpFb + SbSpFa + S-pFbra, 
each of which may evidently be put in the form Syp, where y is a 
known vector. 

Similarly * the term S.apbpc may be reduced to a set of terms, 
each of which has one of the forms 

Ap^, [Sapf, SapSpp, 
the second being merely a particular case of the third. Thus (the 
numerical factors 2 being introduced for convenience) we may write 
the general scalar equation of the second degree as follows : — 

2S.SapS0p + Ap'^ + 2Syp = a (1) 

249.] Change the origin to 1) where OJD = 6, then p becomes 
p + b, and the equation takes the form 
22.SapS0p + Ap^+21(SapSpb + SfipSab) + 2AS&p+2Syp 

+ 2-S.SabSl3b + Ab^ + 2Syb—C=0; 
from which the first power of p disappears, that is tie surface is 
referred to its centre, if 

2(o-S'y38 + ;3<S'a8) + J5 + y = 0, (2) 

• For S.aphpc=S.capip=S.a'php = (2Sa'Sb—Sa'b)p' + 2Sa'p8bp; and in particular 
cases we may have Va'= Vb. 


a vector equation of the first degree^ wLicli in general gives a single 
definite value for 8, by the processes of Chapter V. [It would lead 
us beyond the limits of an elementary treatise to consider the 
special cases in which (2) represents a line, or a plane, any point of 
which is a centre of the surface. The processes to be employed in 
such special cases have been amply illustrated in the Chapter re- 
ferred to.] 

With this value of 6, and putting 

the equation becomes 

2'L.SapSpp + Ap^=I). 

If 2? =^ 0, the surface is conical (a case treated in last Chapter) ; 
if not, it is an ellipsoid or hyperboloid. Unless expressly stated not 
to be, the surface will, when B is not zero, be considered an ellip- 
soid. By this we avoid for the time some rather delicate con- 

By dividing by B, and thus altering only the tensors of the 
constants, we see that the equation of central surfaces of the second 
order, referred to the centre, is (excluding cones) 

2^{Sap8fip)+gp' = \ (3) 

250.] Differentiating, we obtain 

2'S{SadpSfip + SapS^dp} + 2gSpdp = 0, 
or 8.dp{1{a8pp + pSap) +gp} = 0, 

and therefore, by § 137, the tangent plane is 

<S(ot-p) {■2{cuS^p + pSap)+gp} = 0, 
i.e. S.'!!T{l(aSl3p + pSap)+ffp} = 1, by (3). 

Hence,if v = l{aSfip + pSap) + ffp, (4) 

the tangent plane is Svur = 1, 

and the surface itself is Si>p = 1. 

And, as v'^ (being perpendicular to the tangent plane, and satis- 
fying its equation) is evidently the vector-perpendicular from the 
origin on the tangent plane, v is called the vector ofpronmity. 

251.] Hamilton uses for v, which is obviously a linear and vector 
function of p, the notation ^p, expressing a functional operation, 
as in Chapter V. But, for the sake of clearness, we will go over 
part of the ground again, especially for the benefit of students who 
have mastered only the more elementary parts of that Chapter. 

We have, then, (fip z=2{aSpp+^Sap)+ffp. 


With this definition of (f>, it is easy to see that 

(«.) (j>{p + a-) = (f)p + <f>(T, &e., for any two or more vectors. 
(5.) (f) (a;/)) = :e(l>p, a particular case of (a), x being- a scalar, 
(c.) d(f>p = (l>{dp). 

{d.) Scr(^p = l,{SacTSfip + S^<TSap)+ffSp(T = Spcpa; 
or <p is, in this ease, self-conjugate. 
This last property is of great importance. 

252.] Thus the general equation of central surfaces of the second 
degree (excluding cones) may now be written 

Sp4>P=l (1) 

Differentiating, Sdpipp + Spd(j>p = 0, 

which, by applying (c.) and then (d.) to the last term on the left, 

gives 2S^pdp=Q, 

and therefore, as in § 250, though now much more simply, the 
tangent plane at the extremity of p is 

5(^-p)# = 0, 
or Stit^p := Sp(f>p = 1. 

If this pass through A{OA = a), we have 

Saipp = 1, 
or, by (d.), Spcfia = 1, 

for all possible points of contact. 

This is therefore the equation of the plane of contact of tangent 
planes drawn from J. 

253.] To find the enveloping cone whose vertex is A, notice that 

{Sp4>p-l)+j){Sp4>a-lf = 0, 

where p is any scalar, is the equation of a surface of the second 

order touching the ellipsoid along its intersection with the plane. 

If this pass through A we have 

{Sa^a—\)-irp{Sa4,a.+ Vf = 0, 
and p is found. Then our equation becomes 

{Sp^p-l){Sa(j>a-l)—{Sp(j)a—lf = 0, (1) 

which is the cone required. To assure ourselves of this, transfer 
the origin to A, by putting p + a for p. The result is, using {a.) 
and (d.), 

{Sp(l)p+2Sp^a + Sa(j}a—l){Sa(j)a—l) — {Sp(pa + Safl}a-lf = 0, 
or Sptpp {Sacfia — 1 ) — (Sp(j)aY = 0, 

which is homogeneous in Tp^, and is therefore the equation of a 

138 QUATEENIONS. [254. 

Suppose A infinitely distant, then we may put in (1) xa for a, 

where x is infinitely great, and, omitting all but the higher terms, 

the equation of the cylinder formed by tangent lines parallel to a is 

{Sp<Pp—l)Sa<i>a—{8p^af = 0. 

254.J To study the nature of the surface more closely, let us 

find the locus of the middle joints of a system of parallel chords. 

Let them be parallel to a, then, if ot be the vector of the middle 
point of one of them, ^a + xa and isr — xa are simultaneous values of 
p which ought to satisfy (1) of § 252. 

That is S.{'!!y±xa)i^{ts±xa)= \. 

Hence, by {a.) and {d.), as before. 

Surd's + x^Sa<j)a = 1, 

S'ST(l>a=zO (1) 

The latter equation shews that the locus of the extremity of ot, 
the middle point of a chord parallel to a, is a plane through the 
centre, whose normal is (pa ; that is, a plane parallel to the tangent 
plane at the point where OA cuts the surface. And {d.) shews that 
this relation is reciprocal — so that if /3 be any value of w, i. e. be 
any vector in the plane (1), a will be a vector in a diametral plane 
which bisects all chords parallel to /3. The equations of these 
planes are Sw^a = 0, 

S-ai^fi = 0, 
so that if F. ^a^/3 = y (suppose) is their line of intersection, we have 
Sycpa = = Sacj>y, \ 

Sy<t>^ = = Sfi,i>yA (2) 

and (1) gives Sficpa = = Sacp/B. ) 

Hence there is an infinite number of sets of three vectors a, /3, y, 
such that all chords parallel to any one are bisected by the diametral 
plane containing the other two. 

255.] It is evident from § 23 that any vector may be expressed 
as a linear function of any three others not in the same plane, let 
then p = xa+yfi + zy, 

where, by last section, Sa^/3 = Sficpa = 0, 
Satpy = Sycpa = 0, 
Sl3(j)y = Sy<l>l3 = 0. 
And let Sacpa = 1. ) 

S/3ct,l3 = 1, [ 
Sycpy = 1, ) 
so that a, /3, and y are vector conjugate semi-diameters of the surface 
we are engaged on. 


Substituting the above value of p in the equation of the surface, 
and attending to the equations in a, /3, y and to (a.), {b.), and (cL), 
we have Sp<l)p = S{m + i/fi + zy) ^ {osa +yfi + zy), 
= x^ +y2 + z^ = 1 . 

To transform this equation to Cartesian coordinates, we notice that 
X is the ratio which the projection of p on a bears to a itself, &c. 
If therefore we take the conjugate diameters as axes of f, j;, f, and 
their lengths as a, b, c, the above equation becomes at once 

^2 -I- §2 + g2 

the ordinary equation of the ellipsoid referred to conjugate diameters. 
256.] If we write —^^ instead of ^, these equations assume an 
interesting form. We take for granted, what we shall afterwards 
prove, that this halving or extracting the root of the vector func- 
tion is lawful, and that the new linear and vector function has the 
same properties («.), {b.), (c), {d.) (§ 251) as the old. The equation 
of the surface now becomes 

Sp^l,^p = -l, 

or ^^P^P = — 1) 

or, finally, T^p = 1. 

If we compare this with the equation of the unit-sphere 

we see at once the analogy between the two surfaces. TAe sphere 
can be changed into the ellipsoid, or vice versa, by a linear deformation 
of each vector, the operator being the function yjr or its inverse. See 
the Chapter on Kinematics. 

257.] Equations (2) § 254 now become 

Sa\l/^I3= =S\j,a\j/^, &c., (1) 

so that yj/a, \lf^, \(ry, the vectors of the unit-sjahere which correspond to 
semi-conjugate diameters of the ellipsoid, form a rectangular system. 

We may remark here, that, as the equation of the ellipsoid referred 
to its principal axes is a case of § 255, we may now suppose i,j, and 

3tj TJ Hy 

k to have these directions, and the equation is -^ + j^ -^ — 2 = ^j 
which, in quaternions, is 

{SipY {Sjpf {Skpf _ 
Sp<i>P=-^ + -^ + —^- - 1- 

We here tacitly assume the existence of such axes, but in all cases, 
by the help of Hamilton's method, developed in Chapter V, we at 
once arrive at the cubic equation which gives them. 

140 QUATERNIONS. [258. 

It is evident from the last-written equation that 
iSip jSjp kSkp 


a'' b'' c" 

^ V a b ' 

which latter may be easily proved by shewing that 

And this expression enables us to verify the assertion of last section 
about the properties of ■^. 

As 8ip=. —X, &c., x,y, z being the Cartesian coordinates referred 
to the principal axes, we have now the means of at once transform- 
ing any quaternion result connected with the ellipsoid into the or- 
dinary one. 

258.] Before proceeding to other forms of the equation of the 
ellipsoid, we may use those already given in solving a few problems. 

Find ike locus of a point when the perpendicular from the centre on 
its polar plane is of constant length. 

If OT be the vector of the point, the polar plane is 
Spt^T^ = 1, 
and the length of the perpendicular from is ^f- — (§ 208). 

Hence the required locus is 

T4>^ = G, 
or ^OT()!)V=-C2, 

a concentric ellipsoid^ with its axes in the same direction as those 
of the first. By § 257 its Cartesian equation is 

259.] Find the locus of a point whose distance from a given point 
is always in a given ratio to its distance from a given line. 

Let p=xj3 be the given line, and A{OA=a) the given point, and 
let Safi = 0. Then for any one of the required points 

Tip-a) = eTrpp, 
a surface of the second order, which may be written 
p^-2Sap+a^ = e2 (6'2/3p_/3V)- 
Let the centre be at 8, and make it the origin, then 

p^ + 2Sp{b-a) + {b-af = e^S^.^{p + b)-fi^{p-\-by}, 
and, that the first power of p may disappear, 
{b-a) = e^{l3Sl3b-l3^), 
a linear equation for 6. To solve it, note that <Sa/3 = 0, operate by 
S.^ and we get (1 -e^/S^ + e^^^)S^b = S^b = 0. 


Hence 8-a = -e^\ 


Referred to this point as origin the equation becomes 

which shews that it belongs to a surface of revolution (of the second 
order) whose axis is parallel to /3, as its intersection with a plane 
S^p = a, perpendicular to that axis, lies also on the sphere 


e^a^ e^/3^a^ 

H-e2/32 {1 + e^^y 
In fact, if the point be the focus of any meridian section of an 
oblate spheroid, the line is the directrix of the same. 

260.] A sphere, jiassing through the centre of an ellipsoid, is cut hy 
a series of spheres whose centres are on the ellipsoid and which pass 
through the centre thereof; find the envelop of the planes of inter- 

Let [p — df = o^ be the first sphere, i.e. 
p^ — 2Sap= 0. 
One of the others is p^ — 2&3-p = 0, 

where Snrcjyss- = 1 . 

The plane of intersection is 

S{7s — a)p = 0. 
Hence, for the envelop, (see next Chapter,) 

S'sr d>nr = 0, ) , , , 

„ , „ > where cr = afar, 
S'urp = 0, ) 

or <^OT = xp, {Vx = 0}, 

i.e. CT = co(l)~'^p. 

Hence x^Sp^-^.p =1, 1 

and xSp<l)~'^p = Sap, ) 

and, eliminating x, 

Sp,j>-^p = {Sap)^ 
a cone of the second order. 

261. J From a point in the outer of two concentric ellipsoids a tan* 
gent cone is d/rawn to the inner, find the envelop of the plane of contact. 
If Si!r(f>zT = 1 be the outer, and iSp^p = 1 be the inner, <f) and -^ 
being any two self-conjugate linear and vector functions, the plane 
of contact is Surxj/p = 1. . 

Hence, for the envelop, Sm'^p = 0, 


tt'^P = 0, ) 

3-'(^CT =: 0, ) 

142 QUATERNIONS. [262. 

therefore (^ot = a!\//p, 

or tn- = x<^~^-\\ip. 

This gives xS.^p(^~'^^p = !> ) 

and x'^S.^p(\>~^-^p = 1, ) 

and therefore, eliminating x, 

or S.p\j/tj)~^-^p = 1, 

another concentric ellipsoid, as \jf(l)~^\jf is a linear and vector func- 
tion = \ suppose ; so that the equation may be written 

Spxp= 1. 
263.] Find the locus of intersection of tangent planes at the extre- 
mities of conjugate diameters. 

If a, /3, y be the vector semi-diameters, the planes are 
8vr]f'^a= — \, •\ 

with the conditions § 257. 

Hence —^^v!S.-^w\i^^y=^'ss = ^a-\--<^^-V'^y, by § 92, 
therefore T^ts = vS, 

since yjra, ^jS, \jfy form a rectangular system of unit- vectors. 
This may also evidently be written 

fci/^^^ = - 3, 
shewing that the locus is similar and similarly situated to the given 
ellipsoid, but larger in the ratio -s/s : 1 . 

263.] ' Find the locus of the intersection of three sjiheres whose dia- 
meters are semi-conjugate diameters of an ellipsoid. 
If a be one of the semi-conjugate diameters 
Sa\l/^a = — 1. 
And the corresponding sphere is 

or p^—S\^ai^~^p = 0, 

with similar equations in /3 and y. Hence, by § 92, 

y}f-^pS.\jra\j/^\j/y = -i'-'^p = p'^{\lfa + \l/l3 + \l/y), 
and, taking tensors, T^'^p = VsTp^, 

or ^-^"^=^3, 

or, finally, Sprj/'^p ;^-3p\ 

This is Fresnel's Surface of Elasticity in the Undulatory Theory. 

264.] Before going farther we may prove some useful properties 
of the function ^ in the form we are at present using — viz. 
iSip jSjp kSkp 



We have p = 

and it is evident that 

(jii =■ 

^J = -i 

# = -^2' 


_ iSip jSjp kSkp 


a* b* C 

<j>~^P = aHSip + bySjp + c^/cSkp, 

and so on. 

Again, if a, /3, y be any rectangular unit-vectors 

But as 

we have 

„2 ^ ^2 
&c. = &c. 

(Sipf + {Sjp)^ + {Skpf=-p\ 

Sa(f,a + Sfi<l>^ + Sy<py = 1^ + ^+1^ 





*.♦,*,=«.(= + ...)(5^ + ...)('2?+...) 



b^ '■ 

b^ ■ 





— 1 


Sia, Sja, Ska 
Sip, Sjl3, Skp 
Siy, Sjy, Sky 

= + 


And so on. These elementary investigations are given here for the 
benefit of those who have not read Chapter V. The student may 
easily obtain all such results in a far more simple manner by means 
of the formulae of that Chapter. 

265.] MnAthe locus of intersection of a rectangular system of three 
tangents to an ellipsoid. 

If tn- be the vector of the point of intersection, a, /3, y the tangents, 
then, since •m + xa should give equal values of a; when substituted in 
the equation of the surface, giving 

S {m + Xa) <p {-or + xa) = 1, 
or x^Sa(\)a + 2xS^(f>a + (/Soti^ct — 1 ) = 0, 

we have {S^ipaY = Sa<l)a {S-sr(j)w—l). 

Adding this to the two similar equations in /3 and y 
(/Sa^ti7)2 + (S/Scp^f + {Sy(l)wf = {Sa^a + <S/3<^/3 + Sy<l>y) (/Stsrc/.w - 1 ), 

144 QUATERNIONS. [266. 

or -{<}>^f = (1 + 1, + ^) {S:^^-l), 

an ellipsoid concentric with the first. 

366.] If a rectangular system of chords he drawn through any point 
within an ellipsoid, the sum of the reciprocals of the rectangles under 
the segments into which they are divided is constant. 

With the notation of the solution of the preceding problem, w 
giving the intersection of the vectors, it is evident that the product 
of the values of x is one of the rectangles in question taken nega- 

Hence the required sum is 

1 £ 1 

Sta^TH — 1 Ssy^'ST — 1 

This evidently depends on Smcfrar only and not on the particular 
directions of a, ^,y : and is therefore unaltered if ■nr be the vector 
of any point of an ellipsoid similar, and similarly situated, to the 
given one. [The expression is interpretable even if the point be 
exterior to the ellipsoid.] 

267.] Shew that if any rectangular system of three vectors he drawn 
from a point of am, ellipsoid, the plane containing their other extremities 
passes through a jimed point. Find the locus of the latter point as tlie 
former varies. 

With the same notation as before, we have 

SsT(j)Zl7 ^ 1, 

and 8 (;sr + X a) (j) (tn- + xa) = 1 ; 

, , „ 2Sa<b-sr 

thereiore x = • 


Hence the required plane passes through the extremity of 

and those of two other vectors similarly determined. It therefore 
passes through the point whose vector is 

aSa^Tjy + ^SlB^yar + ySycjiZT 

Sa(t>a + S^(l)l3 + 8y(l>y ' 

or 6 = ^+^-^ (§173). 

Thus the first part of the proposition is proved. 




But we have also ot = — ("(^ + — ) 
whence by the equation of the ellipsoid we obtain 

the equation of a concentric ellipsoid. 

268.] Find the directions of the three vectors which are parallel to 
a set of conjugate diameters iti each qf two central surfaces of the second 

Transferring the centres of both to the origin, let their equations 
be Sp(t>p.— 1 or 0,; 

and Sp\l/p= 1 or O.S ^' 

If a, l3, y be vectors in the required directions, we must have (§254) 

Sa(p^ = 0, Sa\{/^ = 0, \ 

S^<t>y^Q, S^^lry=Q^ (2) 

Sy(l>a = 0, Syfa = 0. ) 

From these equations 0a || V^y || ^a, &c. 
Hence the three required directions are the roots of 

r.<t>pi'p = o (3) 

This is evident on other grounds, for it means that if one of the 
surfaces expand or contract uniformly till it meets the other, it will 
touch it successively at points on the three sought vectors. 
We may put (3) in either of the following forms — 

or r.p\/f-i(|)p= 0;i ^ '' 

and, as <j) and v/f are given functions, we find the solutions by the 
processes of Chapter V. 

[iVbfe. As (j)~^^ and V~^^ ^^^ ^°^> ^^ general, self-conjugate 
functions, equations (4) do not signify that a, /3, y are vectors parallel 
to the principal axes of the surfaces 

<S.p0-Vp = 1> S.p^jf-^(t)p = 1. 
In these equations it does not matter whether (j)~^^ is self-conjugate 
or not ; but it does most particularly matter when they are differ- 
entiated, so as to find axes, &c.] 

Given two surfaces of the second degree, there exists in general a set 
of Cartesian axes, whose directions are those of conjugate diameters in 
every one of the surfaces of the second degree passing through the inter- 
section of the two surfaces given. 


146 QUATERNIONS. [269. 

For any surface through the intersection of 

Sp(j)p=l and S{p—a)^{p—a) = e, 
is fSp4>p—8{p—a)-f{p — a)=f—e, 

where/ and e are scalars. 

The axes of this depend only on the term 

Hence the set of conjugate diameters which are the same in all are 
the roots of 

J'i/'t>-^)pU'i4>-^)p=0, or rcpp^p=0, 
as we might have seen without analysis. 

The locus of the centres is given by the equation 

('/'-/^)P-V'« = o> 
where/" is a scalar variable. 

269.] Find the equation of the ellipsoid of which three conjugate 
semi-diameters are given. 

Let the vector semi-diameters be a, j3, 7, and let 

8p4,p = 1 
be the equation of the ellipsoid. Then (§ 255) we have 
Sa(i)a = \, Sa(pfi = 0, 

Sycjyy = 1, Sy^a ^ ; 

the six scalar conditions requisite (§ 139) for the determination of 
the linear and vector function (j). 
They give a \\ V(j)^(j)y, 

or xa = (j}~^ F/3y. 

Hence ■ cc = ccScupa = S.afiy, 

and similarly for the other combinations. Thus, as we have 

pS.a^y = cuS.^yp+^S.yap + yS.afip, 
we find at once 

<j)pS^.al3y = Fl3yS.j3yp + VyaB.yap+ FafiS.a^p; 
and the required equation may be put in the form 
S^.afiy = S^.a^p + S^.fiyp + S^.yap. 
The immediate interpretation is that if four tetrahedra be formed iy 
growping, three and three, a set of semi-conjugate vector axes of an 
ellipsoid and any other vector of the surface, the sum of the squares of 
the volumes of three of these tetrahedra is equal to the square of the 
volume of the fourth. 


370.] When the equation of a surface of the second order can be 
put in the form Sp(()-''-p = I, (1) 

where (<^-^)(<#>-^i)(<l'-^2) = 0. _ 

we know that ff, ff^ , g^ are the squares of the principal semi-diameters. 
Hence, if we put (|) + ^ for <^ we have a second surface, the diifer- 
enees of the squares of whose principal semiaxes are the same as for 

thefirst. Thatis, 8p{<^ + h)-'^p=\ (2) 

is a surface confocal with (1). From this simple modification of the 
equation all the properties of a series of confocal surfaces may easily 
be deduced. We give one as an example. 

271.] Any two confocal surfaces of the second order, wJiich meet, 
intersect at right angles. 

For the normal to (2) is, evidently, 

and that to another of the series, if it passes through the common 
point whose vector is p, is there 

(<^ + /ii)-V. 

But ^.(<^+^)-v(^+^o-P = ^•P (^^,)(^^^y 

and this evidently vanishes if h, and h-^ are different, as they must be 
unless the surfaces are identical. 

272.] To find the conditions of similarity of two central surfaces 
of the second order. 

Referring them to their centres, let their equations be 

8p<^'p=\.\ ^'^ 

Now the obvious conditions are that the axes of the one are pro- 
portional to those of the other. Hence, if 

g^-\-m^g'^ + m^g ^m=fi,\ 

^g'nm\g'+m'=0,i ^'^ 

/' + 

be the equations for determining the squares of the reciprocals of 
the semiaxes, we must have 

—^=IJ; -^ = IJ.^, — = IJ,^, (3) 

m^ m^ m ' 

where \x. is an undetermined scalar. Thus it appears that there are 
but two scalar conditions necessary. Eliminating jn we have 

ni'\ _ nn'y m'm\ _ m'\ 

m% ~ %' mm^ ~ mf ^ ■' 

which are equivalent to the ordinary conditions. 

L a 

148 QUATERNIONS. [273. 

273.] Find. the greatest and least semi-diameters of a central plane 

sectioti of an ellipsoid. 

Here Spcl,p = I I 

Sap=o] ^ > 

together represent the elliptic section ; and our additional condition 

is that Tp is a maximum or minimum. 

Differentiating the equations of the ellipse, we have 

S(f>pdp = 0, 

Sadp = 0, 

and the maximum condition gives 

dTp = 0, 

or Spdp = 0. 

Eliminating the indeterminate vector dp we have 

S.apcf>p = (2) 

This shews that tAe maximum or minimum' vector, the normal at its 

extremity, and the perpendicular to the plane of section, lie in one 

plane. It also shews that there are but .two vector-directions which 

satisfy the conditions, and that they are perpendicular to each other, 

for (2) is satisfied if ap be substituted for p. 

We have now to solve the three equations (1) and (2), to find the 

vectors of the two (four) points in which the ellipse (1) intersects 

the cone (2). We obtain at once 

4>p = xV.<^~'^dVap. 

Operating by S.p we have 

1 = xp^Sa(l)~^a. 

XT 2 J. Sp(j)-''-a 

Hence p'op = p-a „ , ^ 

»' '=^('-''«--> « 

fromwhich ■ S.a{l—p^(f))-^a= ; (4) 

a quadratic equation in p^, from which the lengths of the maximum 
and minimum vectors are to be determined. By § 147 it may be 
written mp*Sa(l)-'^a—p^S.a{m2—(t>)a+a' = (5) 

[If we had operated' by 8.<p-^a or by 8.(pr^p, instead of by S.p, 
we should have obtained an equation apparently different from this, 
but easily reducible to it. To prove their identity is a good exercise 
for the student.] 

Substituting the values of p^ given by (5) in (3) we obtain the 
vectors of the required diameters. [The student may easily prove 
directly that {\—pl<f>)-'^a and {l—pl^)-^a 


are necessarily perpendicular to each other, if both be perpendicular 
to a, and if pf and p| be different. See § 271.] 
274.] By (5) of last section we see that 
2 2 _ "^ 

Hence the area of the ellipse (1) is 

V — mSa<f)~^a 
Also the locus of normals to all diametral sections of an ellipsoid, 
whose areas are equal, is the cone 

Sa(t>-'^a = Co?. 
When the roots of (5) are equal, i.e. when 

{m..fl^—Sa^af = ima'^Satp-'^a, (6) 

the section is a circle. It is not difficult to prove that this equation 
is satisfied by only two "Values of Ua, but another quaternion form 
of the equation gives the solution of this and similar problems by 
inspection. (See § 275 below.) 

275.] By § 168 we may write the equation 

Sp<f>p =: 1 

in the new form S.Kpfxp + pp^ = 1, 

where ^ is a known scalar, and A. and f/. are definitely known (with 
the exception of their tensors, whose product alone is given) in 
terms of the constants involved in </>. [The reader is referred again 
also to §§ 121, 122.] This may be written 

2SkpSij.p + {p—SKiJ.)p^ = l (1) 

From this form it is obvious that the surface is cut by any plane 
perpendicular to A. or fi in a circle. For, if we put 

S\p = a, 
we have 2aSixp + {p—S\ix)p^ = 1, 

the equation of a sphere which passes through the plane curve of 

Hence X and n o( § 168 are the values of a in equation (6) of the 
preceding section. 

276.] Any two circular sections of a central surface of the second 
order, whose planes are not parallel, lie on a sphere. 

For the equation {S\p—a) (Sixp — b) = 0, 
where a and b are any scalai* constants whatever, is that of a 
system of two non-parallel planes, cutting the surface in circles. 
Eliminating the product SKpS^p between this and equation (1) of 
last section, there remains the equation of a sphere. 

150 QUATEENIONS. [277. 

277.] To find the generating lines of a central surface of the second 

Let the equation be Spcpp = 1 ; 

then, if a be the vector of any point on the surface, and ■nr a vector 
parallel to a generating line, we must have 

p = a + xm 
for all values of the scalar x. 

Hence 8 {a + xw) <^ (o + xm) = 1 , 

which gives the two equations 


The first is the equation of a plane through the origin parallel to 
the tangent plane at the extremity of a, the second is the equation 
of the asymptotic cone. The generating lines are therefore parallel 
to the intersections of these two surfaces, as is well known. 
Froni these equations w.e have 

ycfysT = Fota- 
where _^ is a scalar to be determined. Operating on this by S.^ and 
S.y, where y3 and y are any two vectors not coplanar with a, we have 

S^{ycl>^+ra^) = 0, Sm{i/<t,y—rya) = (1) 

Hence S.<})a (j^^^ + Fa/3) {y(j)y— Vya) = 0, 

or my^S.a^y—SacpaS.a^y = 0. 

Thus we have the two values 

Sa<f>'sr = 0, 

a /I 

belonging to the two generating lines. 

278.] But by equation (1) we have 

zm = r.(y^/3+ Va^) {y^y— Vyd) 

= my"^ (j)-^ V^y + yV.^a V^y — aS.aVfiy ; 
which, according to the sign of y, gives one or other generating 

Here V^y may be any vector whatever, provided it is not per- 
pendicular to a (a condition assumed in last section), and we may 
write for it 6. 

Substituting the value of y before found, we have 

zvT = (t)-^d—ajSa0 + ^ — Fd>a0, 


or, as we may evidently write it, 

= <i>-'^{r.ar4>ae)±J~r^ae (2) 

Put r = V^a6, 

and we have zur = d>-^ Far + ^— t, 

~ ^ m 
with the condition Srcpa = 0. 

[Any one of these sets of values forms the complete solution of the 

problem ; but more than one have been given, on account of their 

singular nature and the many properties of surfaces of the second 

order which immediately follow from them. It will be excellent 

practice for the student to shew that 

is an invariant. This may most easily be done by proving that 

V.y^e-^Oi = identically.] 
Perhaps, however, it is simpler to write a for F/3y, and we thus 

«CT- = — d) '■ya yaAa + x/ — Va<i>a. 
^ m 

[The reader need hardly be reminded that we are dealing with the 

general equation of the central surfaces of the second order — the 

centre being origin.] 


1 . Find the locus of points on the surface 

Sp<^p = 1 
where the generating lines are at right angles to one another. 

2. Find the equation of the surface described by a straight line 
which revolves about an axis, which it does not meet, but with 
which it is rigidly connected. 

3. Find the conditions that 

Sp^p = 1 
may be a surface of revolution, with axis parallel to a given vector. 

4. Find the equations of the right cylinders which circumscribe 
a given ellipsoid. 

5. Find the equation of the locus of the extremities of perpen- 
diculars to central plane sections of an ellipsoid, erected at the 


centre, their lengths being the principal semi-axes of the sections. 
[Fresnel's Wave-Surface. See Chap. XI.] 

6. The cone touching central plane sections of an ellipsoid, which 
are of equal area, is asymptotic to a confocal hyperboloid. 

7. Find the envelop of all non-central plane sections of an ellip- 
soid whose area is constant. 

8. Find the locus of the intersection of three planes, perpendicular 
to each other, and touching, respectively, each of three confocal 
surfaces of the second order. 

9. Find the locus of the foot of the perpendicular from the centre 
of an ellipsoid upon the plane passing through the extremities of a 
set of conjugate diameters. 

10. Find the points in an ellipsoid where the inclination of the 
normal to the radius-vector is greatest. 

1 1 . If four similar and similarly situated surfaces of the second 
order intersect, the planes of intersection of each pair pass through 
a common point. 

12. If a parallelepiped be inscribed in a central surface of the 
second degree its edges are parallel to a system of conjugate dia- 

13. Shew that there is an infinite number of sets of axes for which 
the Cartesian equation of an ellipsoid becomes - 

x^-^y'^+z^ = e^. 

14. Find the equation of the surface of the second order which 
circumscribes a given tetrahedron so that the tangent plane at each 
angular point is parallel to the opposite face; and shew that its 
centre is the mean point of the tetrahedron. 

15. Two similar and similarly situated surfaces of the second 
order intersect in a plane curve, whose plane is conjugate to the 
vector joining their centres. 

16. Find the locus of all points on 

Sp(i>p = 1, 
where the normals meet the normal at a given point. 

Also the locus of points on the surface, the normals at which 
meet a given line in space. 

17. Normals drawn at points situated on a generating line are 
parallel to a fixed plane. 

18. Find the envelop of the planes of contact of tangent planes 
drawn to an ellipsoid from points of a concentric sphere. Find the 
locus of the point from which the tangent planes are drawn if the 
envelop of the planes of contact is a sphere. 


19. The sum of the reciprocals of the squares of the perpendiculars 
froiQ the centre upon three conjugate tangent planes is constant. 

20. Cones are drawn, touching an ellipsoid, from any two points 
of a similar, similarly situated, and concentric ellipsoid. Shew that 
they intersect in two plane curves. 

Find the locus of the vertices of the cones when these plane sec- 
tions are at right angles to one another. 

2 1 . Find the locus of the points of contact of tangent planes 
which are equidistant from the centre of a surface of the second 

22. From a fixed point A, on the surface of a given sphere, draw 
any chord AB; let 1/ be the second point of intersection of the 
sphere with the secant £D drawn from any point £ ; and take a 
radius vector AE, equal in length to SB', and in direction either 
coincident with, or opposite to, the chord AD : the locus of S is an 
ellipsoid, whose centre is A, and which passes through B. (Hamilton, 
Elements, p. 227.) 

23. Shew that the equation 

p (e2_ 1) (e + Saa) = (Sapf - 2eSapSa'p + (Sa'pf + (1 -e^) p\ 
where e is a variable (scalar) parameter, and a, a' unit- vectors, repre- 
sents a system of eonfocal surfaces. {Ibid. p. 644.) 

24. Shew that the locus of the diameters of 

Sp<pp = 1 
which are parallel to the chords bisected by the tangent planes to 
the cone Spfp = 

is the cone S.p(f>yjf~''-'(f)p = 0. 

25. Find the equation of a cone, whose vertex is one summit of 
a given tetrahedron, and which passes through the circle circum- 
scribing the opposite side. 

26. Shew that the locus of points on the surface 

Sp<f)p = 1, 
the normals at which meet that drawn at the point p=t!r, is on the 
cone «S'.(/)— ot) (t)w(j)p = 0, 

27. Find the equation of the locus of a point the square of whose 
distance from a given line is proportional to its distance from a 
given plane. 

28. Shew that the locus of the pole of the plane 

Sap = 1, 
with respect to the surface 

Sp(pp = 1, 


is a sphere^ if a be subject to the condition 
Sacl>-^a = C. 

29. Shew that the equation of the surface generated by hnes 
drawn through the origin parallel to the normals to 

Sp^-'^p = 1 
along its lines of intersection with 

Sp{<l> + F]--^P=zi, 

is m^ —^Sm {4, + ky^-m: = 0. 

30. Common tangent planes are drawn to 

2S\pSiJ,p + {p—Skij.)p^ = l, and Tp = k, 
find the value of A that the lines of contact with the former surface 
may be plane curves. What are they, in this case, on the sphere? 
Discuss the case of jo^—S^Xix = 0. 

31. If tangent cones be drawn to 

Sp(j>^P = 1, 
from every point of 'Sp'pp = !> 

the envelop of their planes of contact is 

Sp^^p = 1. 

32. Tangent cones are drawn from every point of 

S{p — a)(j>{p — a.) = n^, 
to the similar and similarly situated surface 

Sp4,p = 1, 
shew that their planes of contact envelop the surface 

{Sa(l)p-lY = n'^Sp(l)p. 

33. Pind the envelop of planes which touch the parabolas 

p = ai^ + pt, p = aT^ + yr, 

where a, (3, y form a rectangular system, and t and t are scalars. 

34. Find the equation of the surface on which lie the lines of 
contact of tangent cones drawn from a fixed point to a series of 
similar, similarly situatedj and concentric ellipsoids. 

35. Discuss the surfaces whose equations are 

SapS^p = Syp, 
and S^ap + S.a^p— \. 

36. Shew that the locus of the vertices of the right cones which 
touch an ellipsoid is a hyperbola. 

37. If oj, Og, ag be vector conjugate diameters of 

Sp(f>p = 1, 
where ^^— %<^^ +%(^— m = 0, 

shew that 2(a^)=--^> 2(Fa, 0,)^= -} S^.(ua,a~= > 

^ ' m V 1 Z^ ^ "12 3 ^ 

and 2 (<ii>af — m^ . 



279.] We have already seen (§31 (l)) that the equations 
p = ct>t = S.a/it), 
and p = (p{i, u) = 1.af{t, u), 
where a represents one of a set of given vectors, and /"a scalar func- 
tion of scalars t and u, represent respectively a curve and a surface. 
We commence the present too brief Chapter with a few of the im- 
mediate deductions from these forms of expression. We shall then 
give a number of examples, with little attempt at systematic devel- 
opment or even arrangement. 

280.] What may be denoted by t and u in these equations is, of 
course, quite immaterial : but in the case of curves, considered 
geometrically, t is most conveniently taken as the length, s, of the 
curve, measured from some fixed point. In the Kinematical in- 
vestigations of the next Chapter t may, with great convenience, be 
employed to denote time. 

281.] Thus we may write the equation of any curve in space as 

P = <i>^, 
where <^ is a vector function of the length, s, of the curve. Of 
course it is only a linem- function when, the equation (as in § 31 {I)) 
represents a straight line. 

283.] We have also seen (§§ 38, 39) that 

^P ^ A. A.' 

is a vector of unit length in the direction of the tangent at the ex- 
tremity of p. 

At the proximate point, denoted by "s + hs, this unit tangent vector 
becomes ^'s + (|)"s 6« + &e. 

156 QUATERNIONS. [283. 

But., because T<})'s = 1, 

we have S.<j)'s (j/'s = 0. 

Hence ij/'s is a vector in the osculating plane of the curve, and per- 
pendicular to the tangent. 

Also, if bd be the angle bet.ween the successive tangents (j/s and 
<f/s + (p"s bs + , we have 

<^ = ^*'" 

so that t&e tensor of <^"s is the reciprocal of the radius of absolute 
curvature at the point s. 

283.] Thus, if OP = (/>« be the vector of any point P of the 
curve, and if C be the centre of curvature at P, we have 

and thus OC = <hs jj- 

cf) s 

is the equation of the locus of the centre of curvature. 

Hence also F.cjj'skj/'s or <f>s^"s 

is the vector perpendicular to the osculating plane ; and 


is the tortuosity of the given curve, or the rate of rotation of its 
osculating plane per unit of length. 

284.J As an example of the use of these expressions let us fin^ 
the curve whose curvature and tortuosity are both constant. 

We have curvature = T^"s = Tp"= c. 

Hence (j)'s(j/'s = p'p"= ca, 

where a is a unit vector perpendicular to the osculating plane. This 

o r fff . ffo ^ Pi Oct ^j ff ff 

pp +p ^=c^— = cc^Up =Cip , 

if Cj represent the tortuosity. 

Integrating we get p'p"- g^p'^^^ (1) 

where /3 is a constant vector. Squaring both sides of this equation, 
we get c2 = cf -/32 - 2 c^Sfip' 

(for by operating with S.p' upon (1) we get +c^ = Sj3p'), 
or Tj3 = ^/c^+cl 


Multiply (1) by p, remembering that 

Tp'= 1, 
and we obtain _ p" = _ q 4- p'^^ 

or, by integration, p = c-^s—pP-\-a, (2) 

where a is a constant quaternion. Eliminating p', we have 

of which the vector part is 

p"— p/32 = —cjsfi— Fafi. 
The complete integral of this equation is evidently 

P = ieos.sT^ + r,sin.sTl3-~{c^sl3+ Faj3), (3) 

f and T] being any two constant vectors. We have also by (2), 

Sfip = CjS + Sa, 
which requires that Sfi^ = 0, Sfirj = 0. 
The farther test, that Tp'=l, gives us 

-1 = Tl3\i^sin\sT^ + r,^cos\sT^-2Sir,sm.sTl3eos.sTl3)- -/^ • 
This requires^ of course^ 

so that (3) becomes the general equation of a helix traced on a right 
cylinder. (Compare § 31 (m).) 

285.] The vector perpendicular from the origin on the tangent 
to the curve p = rf)« 

is, of course, -,Vp'p, or p'Fpp' 

(since p' is a unit vector). 

To find a common property of curves whose tangents are all equi- 
distant from the origin. 

Here TFpp'^z c, 
which may be written —p^—S^pp'=c^ (1) 

This equatiod shews that, as is otherwise evident, every curve on 
a sphere whose centre is the origin satisfies the condition. For ob- 
viously —p^ = c^ gives Spp'= 0, 
and these satisfy (1). 

If Spp' does not vanish, the integral of (1) is 

VTp^-c^ = s, (2) 

an arbitrary constant not being necessary, as we may measure s 
from any point of the curve. The equation of an involute which 
commences at this assumed point is 

-ST = p — sp'. 

158 QUATERNIONS. [286. 

This gives T^^ = Tp^ + s^ + 2 sSpp' 

= Tp^^s''-2s^/Tp^-c^, by(l), 
= o\ by (2). 
This includes all curves whose involutes lie on a sphere about the origin. 
286.] Find the locus of the foot of the ^perpendicular drawn to a 
tangent to a right helix from a point in the axis. 
The equation of the helix is 

p = acos- +/3sin- 4-y*, 
a a ' 

where the vectors a, ^, y are at right angles to each other, and 

Ta = Tl3=z h, while aTy = ^a^-h^. 

The equation of the required locus isj by last section, 

■ar = p'Vpp' 

, s a^—l^ . *\ ^/ . s a'^—W' S\ b^ 

= a (cos — I 5 — ssin-) + fl(sm 5 — «cos-) + y-^-*. 

^ a a^ a^ ^ a a^ a' ' a^ 

This curve lies on the hyperboloid whose equation is 

B'^aTn-^-S^^vs-a^S^yw = «*, 

as the reader may easily prove for himself. 

287.] To find the least distance between consecutive tangents to a 
tortuous curve. 

Let one tangent be ct = p + xp' , 
then a consecutive one, at a distance hs along the curve, is 

^ = p + p'6« + p"g +&c.+y(p' + /'85 + p"'g +...). 

The magnitude of the least distance between these lines is, by 

^.(p'8* + p"g+p'"j^+...)C^r.p'(p' + p"6* + p"'g + ...) 

~ Trp'p"is 

if we neglect terms of higher orders. 

It may be written, since p'p" is a vector, and Tp' = 1 , 

But (§133(2)) ^^^ = r^5s=p,p'S.p'py' 
Hence pj-,8.Up"rp'p"' 


is the small angle, 6</), betwee:rtlie two successive positions of the 
osculating plane. [See also § 283.] 

Thus the shortest distance between two consecutive tangents is 
expressed by the formula bcfibs^ 

12/ ' 
vhere r, = -=y-, , is the radius of absolute curvature of the tortuous 

288.] Let us recur for a moment to the equation of the parabola 
(§31(/.)) ^ /3<2 

P = "'^ + 2 ■ 
Here p'= {a + fit)-j-, 

whence, if we assume Safi = 0, 

from which the length of the are of the curve can be derived in 
terms of t by integration. 

Again, p"=(a+,0£+K|)^- 

dH _ d 1 _ dt S.^{a + pi) 

^ ds^ ~ ds ' T{a +/3i!) ~ "•" S T{a + ^tf ' 

and therefore, for the vector of the centre of curvature we have 
(§ 283), ^^^f_^§^ -{a^ + ^H^f{-^o? + afiH)-\ 

which is the quaternion equation of the evolute. 

289.] One of the simplest forms of the equation of a tortuous 
curve is fl^2 yp 

P = -i + '^ + \' 

where a, /3, y are any three non-coplanar vectors, and the numerical 
factors are introduced for convenience. This curve lies on a para- 
bolic cylinder whose generating lines are parallel to y ; and also on 
cylinders whose bases are a cubical and a semi-cubical parabola, 
their generating lines being parallel to ^ and a respectively. We 
have by the equation of the curve 


from which, by 2'/=!, the length of the curve can be fouud In 
terms of t ; and 

from which p" can be expressed in terms of s. The investigation 
of various properties of this curve is very easy, and will be of great 
use to the student. 

{Note. — It is to be observed that in this equation t cannot stand 
for *, the length of the curve. It is a good exercise for the student 
to shew that such an equation as 

or even the simpler form 

p- a^ + ^s^, 
involves an absurdity.] 

290.] The equation p = <f)^€, 

where cf) is a. given self-conjugate linear and vector function, t a 
scalar variable, and e an arbitrary vector constant, belongs to a 
curious class of curves. 

We have at once — = ^' log (jbe, 

where \og<p is another self-conjugate linear and vector function, 
which we may denote by x- These functions are obviously commu- 
tative, as they have the same principal set of rectangular vectors, 
hence we may write gp 

which of course gives -j^ = x^Pt &c., 

since x does not involve t. 

As a verification, we should have 

./,-"e = p+^a^ + ^— + &C. 

= (i + s^x+|Jx^+ )p 

where e is the base of Napier's Logarithms. 
This is obviously true if ^" = e*''', 
or (jb = gXj 

or log = X. 
which is our assumption. 

[The above process is, at first sight, rather startling, but the 


student may easily verify it by writing, in accordance with the 
, ilts of Chapter V, 

whence ^«e z= —g[aSaf—gl^^S^e—glySye. 

He will find at once 

X« = —logg^^aSat - hgg^pSfie-loggsySye, 
and the results just given follow immediately.] 

291.] That the equation 

p = (^ (i5, w) = 2 . af{t, u) 
represents a surface is obvious from the fact that it becomes the 
equation of a definite curve whenever either t ov u has a particular 
value assigned to it. Hence the equation at once furnishes us with 
two systems of curves, lying wholly on the surface, and such that 
one of each system can, in general, be drawn through any assigned 
point on the surface. Tangents drawn to these curves at their 
point of intersection must^ of course, lie in the tangent plane, whose 
equation we have thus the means of forming. 

292.] By the equation we have 

* = (§)*+(£)*' 

where the brackets are inserted to indicate partial differential coefii- 
cients. If we write this as 

dp = (ji'f df + (j)\ du, 
the normal to the tangent plane is evidently 

and the equation of that plane 


293.] As a simple example, suppose a straight line to move along 

a fixed straight line, remaining always perpendicular to it, while 

rotating about it through an angle proportional to the space it has 

advanced ; the equation of the ruled surface described will evidently 

be p = at+u(PGOst + ysmt), fl) 

where a, j8, y are rectangular vectorSj and 

T0 = Ty. 
This surface evidently intersects the right cylinder 

p = a (p cos t + y sin t) + va, 
in a helix (§§ 31 (m), 284) whose equation is 

p = o^ + a(/3cos^ + ysini!). 
These equations illustrate very well the remarks made in §§ 3 1 (^, 29 1 


162 QUATERNIONS. [294. 

as to the curves or surfaces represented by a vector equation ac- 
cording as it contains one or two scalar variables. 

From (1) we have 

dp = \a—u{^svo.t—y<iOst)'\dt-\-{^ cos t-\-y&mt)du, 
so that the normal at the extremity of p is 

Ta {y cost-p sin t) - uT^^ Ua. 
Hence, as we proceed along a generating line of the surface, for 
which t is constant, we see that the direction, of the normal changes. 
This, of coursCj proves that the surface is not developable. 

294.] Hence the criterion for a developable surface is that if it 
be expressed by an equation of the form 

p = <f)t-\- tixj/t, 

where (j)t and \jft are vector functions, we must have the direction of 
the normal 'F{<t)'t + wft} \j/t 

independent of u. 

This requires either F-fi-^'t = 0, 
which would reduce the surface to a cylinder^ all the generating 
lines being parallel to each other ; or 

F(j>'t\}/t = 0. 
This is the criterion we seek, and it shews that we may write, for a 
developable surface in general, the equation 

p = (pt + U(p't • (1) 

Evidently p = ^t 

is a curve (generally tortuous) and (f/t is a tangent vector. Hence 

a developable surface is the locus of all tangent lines to a tortuous 


Of course the tangent plane to the surface is the osculating plane 
at the corresponding point of the curve ; and this is indicated by 
the fact that the normal to (1) is parallel to 
r(i>tcl>"t. (See § 283.) 

To find the form of the section of the surface made iy a normal plane 
through a point in the curve. 

The equation of the surface is 

OT = p+«/3' + — p" + &c.+«(p'+*p'' + &c.). 
The part of tsr— p which is parallel to p' is 

-p'^(^-p)p'=-/(-(s+«^)-p"^(4+^) + ...); 

therefore ^-p = Ap'+(~+ws) p"-(~ +'^) p'FpY' + ... . 


And, when A = 0, i.e. in the normal section, we have approximately 

so that ^ _ p = _ i_ p" _ 1_ p' fp'p'". 

Z o 

Hence the curve has an equation of the form 

a semicubical parabola. 

395. J A Geodetic line is a curve drawn on a surface so that its 
osculating plane at any point contains the normal to the surface. 
Hence, if v be the normal at the extremity of p, p and p" the first 
and second differentials of the vector of the geodetic, 

S.vp'p"= 0, 
which may be easily transformed into 

V.vdUp'^ 0. 
296.] In the sphere Tp = ayte, have 

V Up, 
hence S.pp'p"= 0, 

which shews of course that p is confined to a plane passing through 
the origin, the centre of the sphere. 

For a formal proof, we may proceed as follows — 
The above equation is equivalent to the three 

S9p = 0, Sdp'= 0, Sdp"= 0, 
from which we see at once that 5 is a constant vector, and therefore 
the first expression, which includes the others, is the complete in- 

Or we may proceed thus — 

= -pS.ppY+p"s.py= r. Vpp'rpp"= r. Vpp'dVpp', 

whence by § 133 (2) we have at once 

UVpp'= const. = suppose, 
which gives the same results as before. 
297.] In any cone we have, of course, 
Svp = 0, 
since p lies in the tangent plane. But we have also 

Svp'= 0. 
Hence, by the general equation of § 295, eliminating v we get 
= S.pp'rp'p"= SpdUp' by § 133 (2). 

Integrating C=Sp Up'-jsdp Up'= Sp Up' +J Tdp. 

The interpretation of this is, that the length of any arc of the geo- 
detic is equal to the projection of the side of the cone (drawn to its 

164 QTJATEENIONS. [298. 

extremity) upon the tangent to the g«odetic. In other words, when 
the cone is developed the geodetic becomes a straight line. A similar 
result may easily be obtained for the geodetic lines on any develop- 
able surface whatever. 

298.] To find the shortest line connecting two points on a given 

Here / Tdp is to be a minimum, subject to the condition that dp 
lies in the given surface. 
Now h^Tdp = fbTdp = -f^^^ = -fs. Udpdbp 

= - [_S. Udp 8/)] + fs.bpdUdp, 

where the term in brackets vanishes at the limits, as the extreme 
points are fixed, and therefore 8p = 0. 
Hence our only conditions are 


' S.bpdUdp = 0, and Svbp = 0, giving 
V.vdVdp = 0, as in § 295. 
If the extremities of the curve are not given, but are to lie on 
given curves, we must refer to the integrated portion of the ex- 
pression for the variation of the length of the arc. And its form 

shews that the shortest line cuts each of the given curves at right 

299.] The osculating plane of the curve 


is S.4,'t<i,"t{m-p) = 0, (1) 

and is, of course, the tangent plane to the surface 

p = <t)t + U(t>'t (2) 

Let us attempt the converse of the process we have, so far, pursued, 
and endeavour to find (2) as the envelop of the variable plane (1). 
Differentiating (1) with respect to t only, we have 

By this equation, combined with (1), we have 


or zT = p + u(l)'= (l)+'U(l/, 

which is equation (2). 

300.] This leads us to the consideration of envelops generally, 
and the process just employed may easily be extended to the problem 


of finditiff the envelop of a series of surfaces whose equation contains 
one scalar parameter. 

When the given equation is a scalar one^ the process of finding 
the envelop is precisely the same as that employed in ordinary 
Cartesian geometry, though the work is often shorter and simpler. 

If the equation be given in the form 
p =-\}i{t, u, v), 
where t/^ is a vector function, t and u the scalar variables for any 
one surface, v the scalar parameter, we have for a proximate surface 

Pi = V' {h> %. ^i) = p+'Vt^t + 'Vu^'^'^Vv^'"- 
Hence at all points on the intersection of two successive surfaces 
of the series we have 

which is equivalent to the following scalar equation connecting the 
quantities t, u, and v ; 

This equation, along with 

p, = -f{i, u, v), 
enables us to eliminate t, u, v, and the resulting scalar equation 
is that of the required envelop. 

301.] As an example, let us find the envelop of the osculating 
plane of a tortuous curve. Here the equation of the plane is (§ 299), 

S.{m-p)<i/t<i>"t= 0, 
or CT = (l>t+x^'t+i/^"t = •^{x,y, {), 
if p = <f)t 

be the equation of the curve. 

Our condition is, by last section, 

or S.<i>'t 4>"t l(t)'t + so4>"t + y ^'"t] = 0, 
or y84't<^"t<^"'t=(i. 
Now the second factor cannot vanish, unless the given curve 
be plane, so that we must have 

and the envelop is 'si =■ <pt + w<^'t 

the developable surface, of which the given curve is the edge of 

regression, as in § 299. 

302.] When the equation contains two scalar parameters its 
differential coefiieients with respect to them must vanish, and we 
have thus three equations from which to eliminate two numerical 

166 QUATERNIONS. [303. 

A very common form in whieli these two parameters appear ia 
quaternions is that of an unknown unit-vector. In this case the 
problem may be thus stated — Find the envelop of the surface whose 
scalar equation is Jpu^ a) = 0, 

wJiere a is subject to the one condition 

Ta = 1. 

Differentiating with respect to o alone, we have 
Svda = 0, Sada = 0, 

where v is a known vector function of p and a. Since da may have 
any of an infinite number of values, these equations shew that 

Fav = 0. 
This is equivalent to two scalar conditions only, and these, in addi- 
tion to the two given scalar equations, enable us to eliminate a. 

With the brief explanation we have given, and the examples 
which follow, the student will easily see how to deal with any other 
set of data he may meet with in a question of envelops. 

303.] Find the envelop of a plane whose distance from the origin is 

Here Sap =-—c, 

with the condition Ta = 1 . 

Hence, by last section, Vpa = 0, 
and therefore p = ca, 

or Tp = c, 
the sphere of radius c, as was to be expected. 

If we seek the envelop of those only of the planes which are parallel 
to a given vector /3, we have the additional relation 

Sa^ = 0. 

In this case the three differentiated equations are 
Spda = 0, Sada = 0, SjSda = 0, 

and they give S.a^p = 0. 

Hence a = U.^T^p, 

and the envelop is TVfip = cTfi, 

the circular cylinder of radius c and axis coinciding with fi. 

By putting Safi = e, where e is a constant different from zero, 
we pick out all the planes of the series which have a definite in- 
clination to j8, and of course get as their envelop a right cone. 

304.] The equation S'^ap+tS.a^p = h 
represents a parabolic cylinder, whose generating lines are parallel 
to the vector aFa/S. For the equation is of the second degree, and 


is not altered by increasing p by the vector xaFa^ ; also the surface 
cuts planes perpendicular to a in one line, and planes perpendicular 
to FajS in two parallel lines. Its form and position of course depend 
upon the values of a, /3, and 6. It is required to find its envelop if ^ 
and b be constant, and a be subject to the one scalar condition 

The process of § 302 gives, by inspection, 
pSap+ Vfip = oca. 
Operating by S.a, we get 

S^ap + S.aj3p =—«!, 
which gives S.a/Sp = x-i- i. 

But, by operating successively by S. Fj3p and by S.p, we have 

{FPpf = (vS.aISp, 
and {p^—x)Sap = 0. 

Omitting, for the present, the factor Sap, these three equations give, 
by elimination of x and a, 

{rppf = p^{p^+b), 

which is the equation of the envelop required. 

This is evidently a surface of revolution of the fourth order whose 
axis is /3 ; but, to get a clearer idea of its nature, put 

and the equation becomes {V^taf = c* + 6zt^, 
which is obviously a surface of revolution of the second degree, 
referred to its centre. Hence the required envelop is the reciprocal 
of such a surface, in the sense that t^e rectangle under the lengths of 
condirectional radii of the two is constant. 

We have a curious particular case if the constants are so related 
that b + ^^ =zQ, 

for then the envelop breaks up into the two equal spheres, touching 
each other at the origin, P^ = ± ^^Pi 

while the corresponding surface of the second order becomes the 
two parallel planes S^.^ = + e^. 

305.] The particular solution above met with, viz. 

Sap — 0, 
limits the original problem, which now becomes one of finding the 
envelop of a line instead of a surface. In fact this equation, taken 
in conjunction with that of the parabolic cylinder, belongs to that 
generating line of the cylinder which is the locus of the vertices of 
the principal parabolic sections. 

168 QUATERNIONS. [306. 

Our equations become 2S.al3p = h, 

Sap = 0, 
Ta = I; 
whence Ffip = ica, giving 

^^ 2 

and thence ^^fip = - ', 

so that the envelop is a circular cylinder whose axis is /3. [It is to 
be remarked that the equations above require that 

Sa^ = 0, 
so that the problem now solved is merely that of tke envelop of a 
parabolic cylinder which rotates about its focal line. This discussion 
has been entered into merely for the sake of explaining a peculiarity 
in a former result, because of course the present results can be 
obtained immediately by an exceedingly simple process.] 

306.] The equation SapS.ajip = a^, 
with the condition Ta= I, 

represents a series of hyperbolic cylinders. It is required to find 
their envelop. 

As before, we have pS.app+ F/SpSap = xa, 

which by operating by S.a, S.p, and S. Vfip, gives 

2a^ =—x, 
p^S.afip = xSap, 
Eliminating a and x we have, as the equation of the envelop, 

p^iFjSpf = 4.a*. 
Comparing this with the equations 

and {rppY = -2a^, 
which represent a sphere and one of its circumscribing cylinders, 
we see that, if eondirectional radii of the three surfaces be drawn 
from the origin, that of the new surface is a geometric mean be- 
tween those of the two others. 

307.] Find the envelop of all spheres which touch one given line 
and have their centres in another. 

Let p = ^-\-yy 

be the line touched by all the spheres, and let xa be the vector of 
the centre of any one of them, the equation is (by § 200, or § 201) 
y'^ip-xaf =-{r.y{fi-xa)Y, 


ov, putting for simplicityj but without loss of generality, 

Ty=l, Sa^ = 0, iSl3y = 0, 
so that /3 is the least vector distance between the given lines, 

{p—xa)^ = {^—xa)^-\-x'^S'^ay, 
and, finally, P^-fi^- ix Sap = x^ S^ay. 

Hence, by § 300, —2Sap = 2xS^ay. 

[This gives no definite envelop if 

Say = 0, 
i. e. if the line of centres is perpendicular to the line touched by all 
the spheres.] 

Eliminating x, we have for the equation of the envelop 

which denotes a surface of revolution of the second degree, whose 
axis is a. 

Since, from the form of the equation, Tp may have any magnitude 
not less than T^, and since the section by the plane 

Sap = 
is a real circle, on the sphere 

the surface is a hyperboloid of one sheet. 

[It will be instructive to the student to find the signs of the 
values of ^1,^2) ffs ^^ i^ § ^^^j ^^^ thence to prove the above con- 

308.] As a final example let us find the envelop of the hyperbolic 
cylinder SapS^p—o = 0, 

where the vectors a and /3 are subject to the conditions 

Ta = T^^ 1, 
Say = 0, aS^8 = 0, 

y and 6 being given vectors. 

[It will be easily seen that two of the six scalars involved in a, /3 
still remain as variable parameters.] 

We have Sada = 0, Syda = 0, 

so that da = xVay. 

Similarly ^/3=yFj35. 

But, by the equation of the cylinders, 

SapSpd/S + SpdaSfip = 0, 
or ySapS.^hp +xS.aypSfip = 0. 

Now by the nature of the given equation, neither Sap nor S^p can 
vanish, so that the independence of da and d^ requires 
S.ayp = 0, S.fibp = 0. 

170 QUATEE,]SriONS. [309- 

Hence a = U.y Fyp, fi =U.h Ftp, 

and the envelop is T.FypFbp — cTyb = 0, 

a surface of the fourth order^ which may be constructed by laying 
off mean proportionals between the lengths of condirectional radii 
of two equal right cylinders whose axes meet in the origin. 

309.] "We may now easily see the truth of the following general 

Suppose the given equation of the series of surfaces, whose envelop 
is required, to contain m vector, and n scalar, parameters ; and that 
the latter are subject top vector, and q scalar, conditions. 

In all there are 3m +n scalar parameters, subject to 3p + q scalar 

That there may be an envelop we must therefore in general have 
{3m + n) — {3_p + q) = 1, or = 2. 
In the former case the enveloping surface is given as the locus of a 
series of curves, in the latter of a series ot points. 

Differentiation of the equations gives us 3j) + q+l equations, 
linear and homogeneous in the 3m+n differentials of the scalar 
parameters, so that by the elimination of these we have one final 
scalar equation in the first case, two in the second ; and thus in each 
case we have just equations enough to eliminate all the arbitrary 

310.] To find the locus of the foot of the perpendicular drawn from 
the origin to a tangent plane to any surface. 

If Svdp = 

be the differentiated equation of the surface, the equation of the 
tangent plane is S(T!r — p)v=0. 

We may introduce the condition 

Svp = 1, 
which in general alters the tensor of v, so that v~^ becomes the 
required vector perpendicular, as it satisfies the equation 

Smv = 1 . 

It remains that we eliminate p between the equation of the given 
surface, and the vector equation 

The result is the scalar equation (in vr) required. 
For example, if the given surface be the ellipsoid 

^p4>P = 1. 
we have ■sr"^ = v = 4>p, 



so that the required equation is 

or /Sar^-V = OT*, 

which is Fresnel's Surface of Elasticity. (§ 263.) 

It is well to remark that this equation is derived from that of the 
reciprocal ellipsoid Sp(b-''-p = 1 

by putting ot~^ for p. 

3 11. J To find the reciprocal of a given surface with respect to the 
unit sphere whose centre is the origin. 

With the condition 8pv = 1, 

of last section, we see that — u is the vector of the pole of the 
tangent plane S{vT-p)v =(). 

Hence we must put zj=—v, 

and eliminate phj the help of the equation of the given sm-faee. 

Take the ellipsoid of last section, and we have 

so that the reciprocal surface is represented by 

It is obvious that the former ellipsoid can be reproduced from this 
by a second application of the process. 

And the property is general, for 

Spv = 1 
gives, by differentiation, and attention to the condition 

Svdp = 0, 
the new relation Spdv = 0, 

so that p and r are corresponding vectors of the two surfaces : either 
being that of the pole of a tangent plane drawn at the extremity of 
the other. 

312.] If the given surface be a cone with its vertex at the origin, 
we have a peculiar case. For here every tangent plane passes 
through the origin, and therefore the required locus is wbolly at an 
infinite distance. The difficulty consists in Spv becoming in this 
case a numerical multiple of the quantity which is equated to zero 
in the equation of the cone, so that of course we cannot put as above 

Spv = 1. 

313.] The properties of the normal vector v enable us to write 
the partial differentia] equations of families of surfaces in a very 
simple- form. 

Thus the distinguishing property of Cylinders is that all their 


generating lines are parallel. Hence all positions of v must be 
parallel to a given plane — or 

Sav = 0, 
which is the quaternion form of the well-known equation 
,dF dF dF „ 
dx dy dz 

To integrate it, remember that we have always 

Svdp = 0, 
and that as v is perpendicular to a it may be expressed in terms of 
any two vectors, /3 and y, each perpendicular to a. 
Hence v = x^ + yy, 

and xS^dp + ySydp = 0. 

This shews that S^p and Syp are together constant or together 
variable, so that SfSp =f{Syp), 

where/" is any scalar function whatever. 

314.] In Surfaces of Bevolution the normal intersects the axis. 
Hence, taking the origin in the axis a, we have 

S.apv = 0, 
or V = xa + yp. 

Hence xSadp + ySpdp = 0, 

whence the integral Tp =f{Sap). 

The more common form, which is easily derived from that just 
written, is TFap = F{Sap). 

In Cones we have Svp = 0, 

and therefore 

Svdp = S.v{TpdUp+ UpdTp) = TpSvdUp. 
Hence SvdUp = 0, 

so that V must be a function of Up, and therefore the integral is 

AUp) = 0, 

which simply expresses the fact that the equation does not involve 
the tensor of p, i. e. that in Cartesian coordinates it is homogeneous. 

315.] If equal lengths he laid off on the normals drawn to any 
surface, the new surface formed hy their extremities is normal to the 
same lines. 

For we have w = p + a Uv, 

and SvdTn = Svdp + aSvdUv = 0, 

which proves the proposition. 

Take, for example, the surface 

Sp(l>p = 1 ; 


the above equation becomes 

so that ^'=(^ + 

and the equation of the new surface is to be found by eliminating 

~— (written ») between the equations 

1 = <S'.(«(j!)+l)-i,!r<^(a;0+l)-iOT, 



and i=S4 (xcj) + 1 )-^zj(j) {xs^ + 1 y-'^-nr. 

316.] It appears from last section that if one orthogonal surface 
can be drawn cutting a given system of straight lines, an inde- 
finitely great number may be drawn: and that the portions of 
these lines intercepted between any two selected surfaces of the 
series are all equal. 

Let p = a+XT, 

where o- and t are vector functions of p, and x is any scalar, be the 
general equation of a system of lines : we have 

Srdp = = S{p—a)dp 
as the differentiated equation of the series of orthogonal surfaces, if 
it exist. Hence the following problem. 

317.] It is required to find the criterion of integrahility of the 
equation Svdp = (1) 

as the complete differential of the equation of a series of mrfaces. 

Hamilton has given [Elements, p. 702) an extremely elegant solu- 
tion of this problem, by means of the properties of linear and vector 
functions. We adopt a different and somewhat less rapid process, 
on account of some results it offers which will be useful to us in 
the next Chapter ; and also because it will shew the student the 
connection of our methods with those of ordinary differential equa- 
. tions. 

If we assume Fp= C 

to be the integral, and apply to it the very singular operator de- 

Adsed by Hamilton, 

„ . d . d , d 

dx '' dy dz 

^ .dF .dF ,dF 
we have vi^= .^ +^^ +^^- 

174 QUATERNIONS. [3 1 8. 

But p = ix +jy + hz, 

whence dp ■= idx+jdy-\-kdz, 

,^ dF , dF , dF , „,„T, 

and Q = dF=-rdx-\--^dy-ir^rdz——SdpVF. 

dx dy ^ dz '^ 

Comparing with the given equation, we see that the latter repre- 
sents a series of surfaces if p, or a scalar multiple of it, can be ex- 
pressed as VF. 

If v = VF, 

„,-^ ^d^F d'^F d^Fs 
we have ^^ = V^^=-(^ + ^ + ^) ' 

a well-known and most important expression, to which we shall 
return in next Chapter. Meanwhile we need only remark that 
the last-written quantities are necessarily scalars, so that the only 
requisite condition of the integrability of (1) is 

rVv= (2) 

If V do not satisfy this criterion, it may when multiplied by a scalar. 
Hence the farther condition 

rv (wv) = 0, 

which may be written 

FvVw—wrVv = (3) 

This requires that SvVv = (4) 

If then (2) be not satisfied, we must try (4). If (4) be satisfied to 
will be found from (3) ; and in either case (1) is at once integrable. 

[If we put dv = (t>dp 

where </> is a linear and vector function, not necessarily self-con- 
jugate, we have 

rvv=:r(i^ + ...) = rii<t,i+...)=-e, 

by § 173. Thus, if (j) be self-conjugate, e = 0, and the criterion (2) 
is satisfied. If (j) be not self-conjugate we have by (4) for the cri- 
terion Sev= 0. 

These results accord with Hamilton's, lately referred to, but the 
mode of obtaining them is quite difierent from his.] 

3I8.3 As a simple example let us first take lines diverging from, a 
point. Here v\[p, and we see that \i v = p 

Vz; = -3, 
so that (2) is satisfied. And the equation is 

Spdp = 0, 
whose integral Tp ■=■ C 

gives a series of concentric spheres. 


Lines ^perpendicular to, and intersecting, it, fixed line. 
If a be the fixed line, ^ any of the others, we have 
S.a^p = 0, Sa^ =Q, Spdp = 0. 

Here i- \\ aVap, 

and therefore equal to it, because (2) is satisfied. 
Hence S.dpaVap = 0, 

or S.VapFadp = 0, 

whose integral is the equation of a series of right cylinders 

T^rap= C. 
319.] To find the orthogonal trajectories of a series of circles whose 
centres are in, and their planes perpendicular to, a given line. 

Let a be a unit-vector in the direction of the line, then one of 
the circles has the equations 

Tp = G,\ 
Sap = C, 3 
where G and C are any constant scalars whatever. 
Hence, for the required surfaces 

V II d^p II Fap, 
where d^p is an element of one of the circles, v the normal to the 
orthogonal surface. Now let dp be an element of a tangent to the 
orthogonal surface, and we have 

Svdp — S.apdp = 0. 
This shews that dp is in the same plane as a and p, i.e. that the 
orthogonal surfaces are planes passing through the common axis. 

[To integrate the equation S.apdp = 
evidently requires, by § 317^ the introduction of a factor. For 
rvFap = riirai+jVaj + Wak) 
= 2a, 
so that the first criterion is not satisfied. But 
S.FaprVFap = 2S.arap = 0, 
so that the second criterion holds. It gives, by (3) of § 317, 
F.Vu;Fap+2wa = 0, 
or pSaVto — aSpVw + 2 wa = 0. 

That is SaVw = 0, \ 

SpVw = 2w. J 
These equations are satisfied by 

But a simpler mode of integration is easily seen. Our equation 
may be written 

= S.aF^ = Sa^-fi = ^.^alog^Z-^ 
p Up p 

176 QUATERNIONS. [320. 

which is immediately integrable, j3 being an arbitrary but constant 

As we have not introduced into this work the logarithms of ver- 
sors, nor the corresponding angles of quaternions, we must refer to 
Hamilton's Elements for a farther development of this point.] 

320.] To jmd the orthogonal trajectories of a given series of sur- 

If the equation Fp = C, 

give Svdp — f^, 

the equation of the orthogonal curves is 

Vvdp = 0. 
This is equivalent to two scalar differential equations (§ 197), which, 
when the problem is possible, belong to surfaces on each of which 
the required lines lie. The finding of the requisite criterion we 
leave to the student. 

Let the surfaces be concentric spheres. 

Here p^ ^ g. 

and therefore Vpdp= 0. 

Hence Tp^ dUp=-Up Fpdp = 0, 

and the integral is Up = constant, 

denoting straight lines through the origin. 

Let the sv/rfaces be spheres touching each other at a common point. 
The equation is (§ 2 1 8) 

Sap-^ = G, 
whence V.papdp = 0. 

The integrals may be written 

S.aPp = 0, p^+hTVap = 0, 
the first (/3 being any vector) is a plane through the common dia- 
meter; the second represents a series of rings or tores (§323) formed 
by the revolution, about a, of circles touching that ILae at the point 
common to the spheres. 

Let the surfaces be similar, similarly situated, and concentric, sur- 
faces of the second order. 

Here Spxp = C, 

therefore ^XP^P = ^• 

But, by § 290, the integral of this equation is 

p = e'^e 

where (f> and x are related to each other, as in § 290 ; and e is any 
constant vector. 


331 .J To integrate the linear partial differential equation of a 
family of surfaces. 

The equation (see § 3 1 3) 

dx dy dz ~ 

may be put in the very simple form 

S (o-V) V, ■= 0, 

if we write a- = iP+JQ + kB, 

1 „ . d . d , d 

and V=t- — |-;---j.^--. 

dx '' dy dz 

This gives, at once, Vu = t/iFOcr, 
where »« is a scalar and 6 a vector (in whose tensor m might have 
been included, but is kept separate for a special purpose). Hence 
dit = — S{dpV)ii 
= —mS.dddp 
= —S.edr, 
if we put dT = mr.<Tdp 

so that m is an integrating factor of V. (rdp. If a value of m can be 
found, it is obvious, from the form of the above equation, that d 
must be a function of r alone ; and the integral is therefore 

w = F{t) = const, 
where F is an arbitrary scalar function. 

Thus the differential equation of Cylinders is 
' S(dV)u=0, 
where a is a constant vector. Here m=l, and 
M = F{Fap) = const. 
That of Cones referred to the vertex is 
S{pV)u= 0. 
Here the expression to be made integrable is 

But Hamilton long ago shewed that (§133 (2)) 
dUp _ ydp _ V.pdp 
-W~ P~ {Tpf ' 
which indicates the value of m, and gives 
u = F{TJp) — const. 
It is obvious that the above is only one of a great number of 
different processes which may be applied to integrate the differential 
equation. It is quite easy, for instance, to pass from it to the 
assumption of a vector integrating factor instead of the scalar m, 


178 QUATERNIONS. [322. 

aud to derive tlie usual criterion of integrability. There is no diffi- 
culty in modifying the process to suit the case when the right-hand 
member is a multiple of u. In fact it seems to throw a very clear 
light upon the whole subject of the integration of partial differ- 
ential equations. If, instead of S (o-V), we employ other operators 
as S {(tV) S {tV), S.o-VtV, &c. (where V may or may not operate on 
u alone), we can pass to linear partial differential equations of the 
second and higher orders. Similar theorems can be obtained from 
vector operations, as V{<tV)*. 

322.] Find the general equation of surfaces described by a line 
which always meets, at right angles, a fixed line. 

If a be the fixed line, y3 and y forming with it a rectangular unit 
system, then p = a;a +y + zy), 

where y may have all values, but x and z are mutually dependent, 
is one form of the equation. 

Another, expressing the arbitrary relation between x and z is 

But we may also write 

p = aF{x) +ya''P, 

as it obviously expresses the same conditions. 

The simplest case is when F{x) = hx. The surface is one which 
cuts, in a right helix, every cylinder which has a for its axis. 

323.] The centre of a sphere moves in a given circle, find the equa- 
tion of the ring described. 

Let a be the unit-vector axis of the circle, its centre the origin, 
r its radius, a that of the sphere. 

Then [p-^f =-0^ 

is the equation of the sphere in any position, where 

<So/3 = 0, 2)3 = n 
These give S.a^p = 0, and ^ must now be eliminated. The result 
is that ^ = raUVap, 

giving (p^— r^-t-a^)^ = ^r^T'^Vap, 

= 4r^-p^-S^ap), 
which is the required equation. It may easily be changed to 

(p^-a'^ + r^)^ =-4:a^p^-4:rWap, ...: (1) 

and in this form it enables us to give an immediate proof of the 
very singular property of the ring (or tore) discovered by Villarceau. 

* Tait, Proc. R. S. E., 1869-70. 


For the planes S.p (a± ) = 0, 

which together are represented by 

r^{r^-a^)8^ap-a'^S^^p = 0, 
evidently pass through the origin and touch (and cut) the ring. 
The latter equation may be written 

r'^S^ap-a^{8^ap + S^pU^) = 0, 

or r^S^ap + a^{p^ + S^.apU^) =0 (2) 

The plane intersections of (1) and (2) lie obviously on the new 
surface (^2_^2 + y2)2 ^ ia^S^.apUia, 

which consists of two spheres of radius r, as we see by writing its 
separate factors in the form 

(p±aaUpf+r^ = 0. 
334.] It may be instructive to work out this problem from a 
different point of view, especially as it affords excellent practice in 

A circle revolves about an axis passing within it, the perpendicular 
from the centre on the axis lying in the plane of the circle: shew that, 
for a certain position of the axis, the same solid mny he traced out by a 
circle revolving about an external axis in its own plane. 

Let a = •fh'^ + c^ be the radius of the circle, i the vector axis of 
rotation, —ca (where Ta=-\) the vector perpendicular from the 
centre on the axis i, and let the vector 

hi + da 
be perpendicular to the plane of the circle. 
The equations of the circle are 

(p_ca)2 + ^2 + c2 = 0, \ 

S(i + Yia)p = 0. C 

Also —p^ = S^ip + S^ap + S'^.iap, 

= SHp + S^ap+ -^SHp 

by the second of the equations of the circle. But, by the first, 

(/)2 + 5Z)2 = 4c2/SV = -4 {c^p'^+a^SHp), 
which is easily transformed into 

{(?-¥f=-i.a^{p^ + S^ip), 
or p2_52 ^ —2aTrip. 

If we put this in the forms 

p^-h^ = 2aSpp, 
and {p-a^f + c^=:0, 

N a 

180 QUATERNIONS. [32 5- 

where ;3 is a unit-vector perpendicular to i and in tlie plane of i 
and p, we see at once that the surface will be traced out by a circle 
of radius c, revolving about i, an axis in its own plane^ distant a 
from its centre. 

This problem is not well adapted to shew the gain in brevity and 
distinctness which generally follows the use of quaternions ; as, 
from its very nature, it hints at the adoption of rectangular axes 
and scalar equations for its treatment, so that the solution we have 
given is but little different from an ordinary Cartesian one. 

325.] A surface is generated hy a straight line which intersects two 
fixed lines : find the general equation. 

If the given lines intersect, there is no surface but the plane con- 
taining them. 

Let then their equations be, 

p = a + xfi, p = a^ + XiPi- 
Hence every point of the surface satisfies the condition, § 30, 

p=y(a + a;^) + (l-5^)(ai + 3'i^i) (1) 

Obviously y may have any value whatever : so that to specify a 
particular surface we must have a relation between x and x^. By 
the help of this, x^ may be eliminated from (1), which then takes 
the usual form of the equation of a surface 

P = 't>i'«,^)- 
Or we may operate on (1) by F.(a + xj3-- ai—XiJ3i), so that we get 
a vector equation equivalent to two scalar equations (§§ 98, 116), 
and not containing y. From this x and x^ may easily be found in 
terms of p, and the general equation of the possible surfaces may be 
written /"{^t *i) = 0, 

where /" is an arbitrary scalar function, and the values of x and x^ 
are expressed in terms of p. 

This process is obviously applicable if we have, instead of two 
straight lines, any two given curves through which the line must ' 
pass ; and even when the tracing line is itself a given curve, situated 
in a given manner. But an example or two will make the whole 
process clear. 

326.] Suppose the moveable line to le restricted by the condition 
that it is always parallel to a fixed plane. 

Then, in addition to (1), we have the condition 
Sy{a-i^-\-x-yP-^—a — x^) = 0, 
y being a vector perpendicular to the fixed plane. 

We lose no generality by assuming o and Oj, which are any 


vectors drawn from the origin to the fixed lines, to be each per- 
pendicular to y ; for, if for instance we could not assume Sya = 0, it 
would follow that Sy^ = 0, and the required surface would either 
be impossible, or would be a plane, cases which we need not con- 
sider. Hence x^8y^^-x8y^ - 0. 

Eliminating' ajj, by the help of this equation, from (1) of last section, 
we have , „; , , ^ Sy& -. 

Operating by any three non-coplanar vectors and with the charac- 
teristic S, we obtain three equations from which to eliminate a; and y. 
Operating by S.y we find 

Syp = xSjSy. 
Eliminating x by means of this, we have finally 

^■'(« + ^^)(«.+ ^) = «. 

which appears to be of the third order. It is really, however, only 
of the second order, since, in consequence of our assumptions, we 
have Vauj^ \\ y, 

and therefore Syp is a spurious factor of the left-hand side. 

327.] Let the fixed lines he perpendicular to each other, and let 
the moveable line pass through the circumference of a circle, whose 
centre is in the common perpendicular, and whose plane bisects that line 
at right angles. 

Here the equations of the fixed lines may be written 
p = a + x^, p =— a+a?iy, 
where a, j3, y, form a rectangular system, and we may assume the 
two latter to be unit-vectors. 
The circle has the equations 

p^ =—a^, Sap = 0. 
Equation (1) of § 325 becomes 

p = i/{a+xj3} + {l-if){-a + x^y). 
Hence Sar'^p = y—(l—^] = 0, or y = i- 

Also p2= -«2 = (2y-l)2 a'-x^f-xl (1-^)^ 

or 4fl^ = (x^+xl), 
so that if we now suppose the tensors of /3 and y to be each 2 a, we 
may put x = cos 0, x^ = sin 6, from which 

p = (2j^— l)a + y/3cos0+(l— y)ysin5; 

^•^^ ^^""^ {l+Sa-^pf + {l-Sa-^pf = '^ • 

182 QUATBRNIONS. [328. 

For this very simple case the solution is not better than the 
ordinary Cartesian one; but the student will easily see that we 
may by very slight changes adapt the above to data far less sym- 
metrical than those from which we started. Suppose, for instance, 
/3 and y not to be at right angles to one another ; and suppose the 
plane of the circle not to be parallel to their plane, &c., &c. But 
farther, operate on every line in space by the linear and vector 
function (^, and we distort the circle into an ellipse, the straight 
lines remaining straight. If we choose a form of ^ whose principal 
axes are parallel to a, p, y, the data will remain symmetrical, but 
not unless. This subject will be considered again in- the next 

328.] To find the curvature of a normal section of a central surface 
of the second order. 

In this, and the few similar investigations which follow, it will 
be simpler to employ infinitesimals than differentials ; though for a 
thorough treatment of the subject the latter method, as may be seen 
in Hamilton's Elements, is preferable. 

We have, of course, '^/'</>P = Ij 

and, if p + hp be also a vector of the surface, we have rigorously, 
whatever be the tensor ofbp, 

Sip + 8p)<t>{p + bp)= 1. 
Hence 2Sbpcl)p-\-Sbp<j)bp = (1) 

Now </)p is normal to the tangent plane at the extremity of p, so 
that if t denote the distance of the point p + bp from that plane 

i =-SbpU(l)p, 
and (1) may therefore be written 

•itT<i>p-T^bpS.mp^Ubp = 0. 
But the curvature of thfe section is evidently 

"^ T^bp ' 
or, by the last equation, 


In the limit, bp is a vector in the tangent plane ; let ct- be the vector 
semidiameter of the surface which is parallel to it, and the equation 
of the surface gives T^isS .U-stcjjU-st = 1, 

so that the curvature of the normal section, at the point p, in the 
direction of or, is 1 


Hirectly as the perpendicular from the centre on the tangent plane, and 
inversely as the square of the semidiameter parallel to the tangent line, 
a well-known theorem. 

329.] By the help of the known properties of the central section 
parallel to the tangent plane, this theorem gives us all the ordinary 
properties of the directions of maximum and minimum curvature, 
their being at right angles to each other, the curvature in any 
normal section in terms of the chief curvatures and the inclination 
to their planes, &c., &c., without farther analysis. And when, in a 
future section, we shew how to find an osculating surface of the 
second order at any point of a given surface, the same properties 
will be at once established for surfaces in general. Meanwhile we 
may prove another curious property of the surfaces of the second 
order, which similar reasoning extends to all surfaces. 

The equation of the normal at the point p + 8p in the surface 
treated in last section is 

CT- = /3 + 8p+«(^(p + 8/)) (1) 

This intersects the normal at p if (§§ 203, 210) 

S.hp^p^hp = 0, 
that is, by the result of § 273, if 8p be parallel to the maximum or 
minimum diameter of the central section parallel to the tangent 

Let o-j and o-g be those diameters, then we may write in general 
hp =piTi + q(T2, 
where ^ and q are scalars, infinitely small. 

If we draw through a point P in the normal at p a line parallel 
to (Tj, we may write its equation 

OT = p-{-a(j)p+^a^. 
The proximate normal (1) passes this line at a distance (see § 203) 

S . {a(l>p — bp) UF(Ti (t){p + 8/)), 
or, neglecting terms of the second order, 

,,,-p- ■ (op 84pu-i(i)iT-^ + aqS.(l)p<jj(p<T2 + q S.cria^fjyp). 
IT (r-j(pp 

The first term in the bracket vanishes because o-j is a principal vector 

of the section parallel to the tangent plane, and thus the expression 

becomes / a „ \ 

Hence, if we take a — Tel, ^^ distance of the normal from the new 
line is of the second order only. This makes the distance of P from 
the point of contact T(f>pT(Tl, i.e. the principal radius of curvature 

184 QUATERNIONS. [330. 

along the tangent line parallel to o-g. That is, the group of normals 
drawn near a point of a central surface of the second order pass ulti- 
mately through two lines each parallel to the tangent to one principal 
section, and passing through the centre of curvature of the other. The 
student may form a notion of the nature of this proposition by con- 
sidering a small square plate, with normals dravra at every point, 
to he slightly bent, but by different amounts, in planes perpendicular 
to its edges. The first bending will make all the normals pass 
through the axis of the cylinder of which the plate now forms part ; 
the second bending will not sensibly disturb this arrangement, 
except by lengthening or shortening the line in which the normals 
meet, but it will make them meet also in the axis of the new 
cylinder, at right angles to the first. A small pencil of light, with 
its focal lines, presents this appearance, due to the fact that a series 
of rays originally normal to a surface remain normals to a surface 
after any number of reflections and refractions. (See § 315). 

330.] To extend these theorems to surfaces in general, it is only 
necessary, as Hamilton has shewn, to prove that if we write 

dv = (\)dp, 
is a self-conjugate function ; and then the properties of <|), as ex- 
plained in preceding Chapters, are applicable to the question. 

As the reader will easily see^ this is merely another form of the 
investigation contained in § 317. But it is given here to shew 
what a number of very simple demonstrations may be given of 
almost all quaternion theorems. 

The vector v is defined by an equation of the form 
dfp = Svdp, 
where /" is a scalar function. Operating on this by another inde- 
pendent symbol of differentiation, 8, we have 
hdfp = Sbvdp + Svhdp. 
In the same way we have 

dbfp = Sdvhp + Svdbp. 
But, as d and 8 are independent, the left-hand members of these 
equations, as well as the second terms on the right (if these exist 
at all), are equal, so that we have 

Sdvbp = Shvdp. 
This becomes, putting dv = <^dp, 

and therefore Sv = ^6p, 

8bp<pdp = Sdptjibp, 

which proves the proposition. 


331.] If we write the differential of the equation of a surface in 
the form df(t = iSvAp, 

then it is easy to see that 

f{p-\-dp) =fp+2Svdp + Sdvdp + kc., 
the remaining terms containing as factors the third and higher 
powers of Tdp. To the second order, then, we may write, except 
for certain singular points, 

= 2Svdp + Sdvdp, 
and, as before, (§ 328), the curvature of the normal section whose 
tangent line is dp is 1 „ dv 

Yv Tp' 
333.] The step taken in last section, although a very simple one, 
virtually implies that the first three terms of the expansion of 
/(p + dp) are to be formed in accordance with Taylor's Theorem, 
whose applicability to the expansion of scalar functions of quater- 
nions has not been proved in this work, (see § 135); we therefore 
give another investigation of the curvature of a normal section, 
employing for that purpose the formulae of § (282). 
We have, treating dp as an element of a curve, 

Svdp = 0, 
or, making s the independent variable, 

Svp'= 0. 
From this, by a second dififerentiation, 

8^p' + Svp"= 0. 

The curvature is, therefore, since v \\ p" and Tp'— \, 

333.] Since we have shewn that 
dv ^ (f)dp 
where is a self-conjugate linear and vector function, whose con- 
stants depend only upon the nature of the surface, and the position 
of the point of contact of the tangent plane ; so long as we do not 
alter these we must consider if) as possessing the properties explained 
in Chapter V. 

Hence, as the expression for Tp" does not involve the tensor of 
dp, we may put for dp any unit-vector r, subject of course to the 

condition Svt = 0. , (1) 

And the curvature of the normal section whose tangent is r is 

186 QUATERNIONS. [334- 

If we consider the central section of the surface of the second order 

&ss^^-\-Tv = 0, 
made by the plane Svm = 0, 

we see at once that the curvature of the given surface along the normal 
section touched hy t is inversely as the square of the parallel radius in 
the auxiliary surface. This, of course, includes Euler's and other 
well-known Theorems. 

334.J To find the directions of maximum and minimum curvature, 
we have St<^t = max. or min. 

with the conditions^ Svt = 0, 

Tt= 1. 
By differentiationj as in § 273, we obtain the farther equation 

S.VT(\)T = (1) 

If T be one of the two required directions, t'=tUv is the other, for 
the last-written equation may be put in the form 
S.TUv(t>{vTUv) = 0, 
i.e. S.T'<t>{vT') = 0, 

or 8.v/^T = 0. 

Hence the sections of greatest and least curvature are perpendicular to 
one another. 

We easily obtain, as in § 273, the following equation 

S.v{(f)-\-ST^T)-'^V = 0, 

whose roots divided by Tv are the required curvatures. 

335.] Before leaving this very brief introduction to a subject, an 
exhaustive treatment of which will be found in Hamilton's Elements, 
we may make a remark on equation (1) of last section 

S.VT(i)T = 0, 

or, as it may be written, by returning to the no'tation of § 333, 

S.vdpdv = 0. 
This is the general equation of lines of curvature. For, if we define 
a line of curvature on any surface as a line such that normals drawn 
at contiguous points in it intersect, then, bp being an element of 
such a line, the normals 

■ST = p + xv and ■or = p + 5p + y (v + bv) 
must intersect. This gives, by § 203, the condition 

, S.bpvbv = 0, 

as above. 



1 . Find the length of any arc of a curve drawn on a sphere so as 
to make a constant angle with a fixed diameter. 

2. Shew that, if the normal plane of a curve always contains a 
fixed line, the curve is a circle. 

3. Mnd the radius of spherica,l curvature of the curve 

p = (jit. 
Also find the equation of the locus of the centre of spherical 

4. (Hamilton, Bishop Law^ s Premium Examination, 1854.) 

(a.) If p be the variable vector of a curve in space, and if the 
differential Ak be treated as = 0, then the equation 

dT{p-K) = 
obliges K to be the vector of some point in the normal 
plane to the curve. 

(b.) In like manner the system of two equations, where dK 
and d^K are each = 0, 

dT(p-K) = 0, d^T{p-K) = 0, 
represents the axis of the element, or the right line 
drawn through the centre of the osculating circle, per- 
pendicular to the osculating plane. 

(c.) The system of the three equations, in which k is treated 
as constant, 

dT{p-K) = 0, d^T(p-K) = 0, d^T{p-K) = 0, 

determines the vector k of the centre of the osculating 

{d.) For the three last equations we may substitute the follow- 
ing : 

S.{p—K)dp = 0, 

S.{p-K)d\ + dp^ = 0, 
S.{p-K)d^p + 3S.dpd^p = 0. 
(e.) Hence, generally, whatever the independent and scalar 
variable may be, on which the variable vector p of the 
curve depends, the vector k of the centre of the oscu- 
lating sphere admits of being thus expressed : 
3 F.dpd^pS.dpd^p-dp^ F.dpd^p 

K = p + 



(/".) In general, 

d{d-W.dpUp) = d{Tp-^r.pdp) 

= Tp-'^ (sr.pdpS.pdp-p^r.pd^p) ; 

^r.pdpS.pdp-pW.pd^P = p^Tpd{p-^F.dpUp); 
and, therefore, the recent expression for k admits of 
being thus transformed, 

dp*d(dp-^r.d^pUdp ) 
"-P'^ S.d^pd^pUdp 
iff.) If the length of the element of the curve be constant, 
dTdp=0, this last expression for the vector of the centre 
of the osculating sphere to a curve of double curva- 
ture becomes, more simply^ 

K = p + 

or K = p + 

S.dpd^pd^p ' 


{h.) Verify that this expression gives /c = 0, for a curve de- 
scribed on a sphere which has its centre at the origin 
of vectors ; or shew that whenever dTp = 0, d^Tp = 0, 
d^Tp = 0, as well as dTdp = 0, then 

5. Find the curve from every point of which three given spheres 
ajjpear of equal magnitude. 

6. Shew that the locus of a point, the difference of whose dis- 
tances from each two of three given points is constant, is a plane 

7. Find the equation of the curve which cuts at a given angle 
all the sides of a cone of the second order. 

Find the length of any are of this curve in terms of the distances 
of its extremities from the vertex. 

8. Why is the centre of spherical curvature, of a curve described 
on a sphere, not necessarily the centre of the sphere ? 

9. Find the equation of the developable surface whose generating 
lines are the intersections of successive normal planes to a given 
tortuous curve. 

1 0. Find the length of an arc of a tortuous curve whose normal 
planes are equidistant from the origin. 

11. The reciprocals of the perpendiculars from the origin on the 
tangent planes to a developable surface are vectors of a tortuous 


curve ; from whose osculatin^planes the cusp-edge of the original 
surface may be reproduced by the same process. 

12. The equation p=Fa'p, 

where a is a unit- vector not perpendicular to ft represents an ellipse. 
If we put y = Fa^, shew that the equations of the locus of the 
centre of curvature are 

S.pyp = 0, 

Sipp + S^yp = {fiSUapf. 

13. Find the radius of absolute curvature of a spherical conic. 

14. If a cone be cut in a circle by a plane perpendicular to a side, 
the axis of the right cone which osculates it, along that side, passes 
through the centre of the section. 

15. Shew how to find the vector of an umbilicus. Apply your 
method to the surfaces whose equations are 

Spipp = 1, 
and SapS^pSyp = 1. 

16. Find the locus of the umbilici of the surfaces represented by 
the equation Sp {(p + A)-^p=l, 

where A is an arbitrary parameter. 

17. Shew how to find the equation of a tangent plane which 

touches a surface along a line^ straight or curved. Find such planes 

for the following- surfaces 

Spipp = 1, 


and {p^-a'^ + b^y + 4:{a^p^ + 6^S'^ap)= 0. 

18. Find the condition that the equation 

S{p + a)<l>p= 1, 
where ^ is a self-conjugate linear and vector function, may represent 
a cone. 

19. Shew from the general equation that cones and cylinders are 
the only developable surfaces of the second order. 

20. Find, the equation of the envelop of planes drawn at each 
point of an ellipsoid perpendicular to the radius vector from the 

21. Find the equation of the envelop of spheres whose centres lie 
on a given sphere, and which pass through a given point. 

22. Find the locus of the foot of the perpendicular from the 
centre to the tangent plane of a hyperboloid of one, or of two, 


23. 'H.arailtou, Mskqp Law's Premium Hxamination, 1852, 

{a.) If p be the vector of a curve in space, the 'length of the 
element of that curve is Tdp ; and the variation of the 
length of a finite arc of the curve is 

b/Tdp = -fSUdpbdp =-ASUdpbp+/SdUdpbp. 
(5.) Hence, if the curve be a shortest line on a given surface, 
for which the normal vector is v, so that Svbp = 0, this 
shortest or geodetic curve must satisfy the differential 
equation, FvdUdp = 0. 

Also, for the extremities of the arc, we have the limiting 

SUdpo Spo = J SUdp^ 8pi = 0. 
Interpret these results, 
(c.) For a spheric surface, Fvp = 0, pdUdp=Q ; the integrated 
equation of the geodetics is p Udp = ■nr, giving Sxsp = 
(great circle). 
For an arbitrary cylindric surface, 

Sav = 0, adUdp = ; 
the integral shews that the geodetic is generally a helix, 
making a constant angle with the generating lines of 
the cylinder. 
[d.) For an arbitrary conic surface, 

Svp = 0, SpdUdp = ; 
integrate this differential equation, so as to deduce from 
it, TVpUdp = const. 
Interpret this result ; shew that the perpendicular from 
the vertex of the cone on the tangent to a given geo- 
detic line is constant ; this gives the rectilinear develop- 
When the cone is of the second degree, the same property 
is a particular case of a theorem respecting confocal 
(e.) For a surface of revolution, 

S.apv — 0, S.apdUdp = ; 

integration gives, 

const. = S.apUdp = TVapSU (Fap.dp) ; 
the perpendicular distance of a point on a geodetic 
line from the axis of revolution varies inversely as the 
cosine of the angle under which the geodetic crosses a 
parallel (or circle) on the surface. 


(/'.) The diiferential eqrration, S.apdUdp = 0, is satisfied not 
only by the geodeties, but also by the circles, on a 
surface of revolution ; give the explanation of this fact 
of calculationj and shew that it arises from the coinci- 
dence between the normal plane to the circle and the 
plane of the meridian of the surface. 

(g.) For any arbitrary surface, the equation of the geodetic 
may be thus transformed, S.vdpcPp = ; deduce this 
form, and shew that it expresses the normal property 
of the osculating plane. 

(A.) If the element of the geodetic be constant, dTdp = 0, then 
the general equation formerly assigned may be reduced 
to r.vd^p= 0. 
Under the same condition, d^p = —v'^Sdvdp. 

{i.) If the equation of a central surface of the second order 
be put under the form fp = I, where the function _/ 
is scalar, and homogeneous of the second dimension, 
then the diiferential of that function is of the forni 
dfp = 2S.vdp, where the normal vector, v = <l>p, is a dis- 
tributive function of p (homogeneous of the first dimen- 
sion), dv=d(j)p = <l)dp. 
This normal vector v may be called the vector of proximity 
(namely, of the element of the surface to the centre) ; 
because its reciprocal, v~^, represents in length and in 
direction the perpendicular let fall from the centre on 
the tangent plane to the surface. 

(^.) If we make S<T<^p =y(o-, p), this function/" is commutative 
with respect to the tvjo vectors on which it depends, 
f{p, a) =/'(*, p) ; it is also connected with the former 
functiony, of a single vector p, by the relation,/" (p, p) ■=fp : 
so that fp = Sp<pp. 
fdp = Sdpdv ; dfdp = 2S.dv d^p ; for a geodetic, with, con- 
stant element, 

2jdp V 

this equation is immediately integrable, and gives 
const. =Tv-J{fJJdp) = reciprocal of Joachimstal's pro- 
duct, PB. 
(l.) If we give the name of " Didonia" to the curve (discussed 
by Delaunay) which, on a given surface and with a 
given perimeter, contains the greatest area, then for 


such a Didonian curve we have by quaternions the 
formula, fS. Uvdpbp + c h/Tdp = 0, 
where c is an arbitrary constant. 
Derive hence the differential equation of the second order, 
equivalent (through the constant c) to one of the third 
order, g-^Sp = F. UvdUdp. 

Geodeties are, therefore, that limiting case of Didonias for 

which the constant c is infinite. 
On a plane, the Didonia is a circle, of which the equation, 
obtained by integration from the general form, is 

p = ■uT + cUvdp, 
m being vector of centre, and c being radius of circle. 
(m.) Operating by 8. TJdp, the general differential equation of 
the Didonia takes easily the following forms : 
c'-'Tdp =S{Uvdp.dUdp); 
c-^Tdp^ = S{Uvdp.d^p); 
c-'^Tdp^ = S.Uvdpd^p; 


{n.) The vector w, of the centre of the osculating circle to a 

curve in space, of which the element Tdp is constant, 

has for expression, 


Hence for the general Didonia, 

c"i = 5i 


T{p-<^) = cSU'" 


(o.) Hence, the radius of curvature of any one Didonia varies, 
in general, proportionally to the cosine of the inclination 
of the osculating plane of the curve to the tangent 
plane of the surface. 
And hence, by Meusnier's theorem, the difference of the 
squares of the curvatures of curve and surface is con- 
stant J the curvature of the surface meaning here the 
reciprocal of the radius of the sphere which osculates 
in the reduction of the element of the Didonia. 

{p.) In general, for any curve on any surface, if £ denote the 
vector of the intersection of the axis of the element (or 


the axis of the circle osculating to the curve) with the 
tangent plane to the surface, then 

Hence, for the general Didonia, with the same significa- 
tion of the symbols, 

£ = p — cTIvdp ; 
and the constant c expresses the length of the interval 
p— f, intercepted on the tangent plane, between the 
point of the curve and the axis of the osculating 

{q.) If, then, a sphere be described, which shall have its centre 
on the tangent plane, and shall contain the osculating 
circle, the radius of this sphere shall always be equal 
to c. 

[r.) The recent expression for ^, combined with the first form 
of the general differential equation of the Didonia, gives 
di = -crdUv Udp ; Vvd^ = 0. 

(«.) Hence, or from the geometrical signification of the con- 
stant c, the known property may be proved, that if a 
developable surface be circumscribed about the arbitrary 
surface, so as to touch it along a Didonia, and if this 
developable be then unfolded into a plane, the curve 
will at the same time be flattened (generally) into a 
circular arc, with radius = c. 

24. Find the condition that the equation 

may give three real values of y for any given value of p. Ifybe a 
function of a scalar, parameter ^, shew how to find the form of this 
function in order that we may have 

^ ^ dx^ ^ df ^-dz^ 
Prove that the following is the relation between / and ^, 

,.=./• ^f =f^ 

^ ^{9i+f)i9^+f){9z+f) ^ ^^f 
in the notation of § 147. 

25. Shew, after Hamilton, that the proof of Dupin's theorem, 
that "each naember of one of three series of orthogonal surfaces 
cut? each member of each of the other series along its lines of 
curvature," may be expressed in quaternion notation as follows : 


If Svdp = 0, Sv'dp — 0, S.vv'dp = 

be integrable, and if 

Svv'= 0, then Fv'dp = 0, makes S.vv'dv = 0. 

Or, as follows, 

If SvVv = Q, Sv'Vv'=0, Sv"Vv"=:0, and r.w'v"= 0, 

then S.v"{Sv'V.v)=:0, 

1 „ . d . d J d 

where V = i-r-+;T-+«-i-- 

dx dy dz 

26. Shew that the equation 

Vap = pVfip 
represents the line of intersection of a cylinder and cone, of the 
second order, which have /3 as a common generating line. 

27. Two spheres are described, with centres at A, B, where 
OA = a, OB — y3, and radii a, h. Any line, OFQ,, drawn from the 
origin, cuts them in T, Q respectively. Shew that the equation of 
the locus of intersection of AT, BQ has the form 

r{a + aU{p~a)) {fi + bU(p-fi)) = 0. 
Shew that this involves S.a^p = 0, 

and therefore that the left side is a scalar multiple of V.afi, so that 
the locus is a plane curve. 

Also shew that in the particular case 

Fal3 = 0, 
the locus is the surface formed by the revolution of a Cartesian 
oval about its axis. 



336.] When a point's vector, p, is a function of the time t, we 

have seen (§36) that its vector- velocity is expressed by -j- or, in 
Newton's notation, by p. 

That is, if p = cpt 

be the equation of an orbit, containing (as the reader may see) not 
merely the form of the orbit, but the law of its description also, then 

p = ^'t 
gives at once the form of the Hodograph and the law of its de- 

This shews immediately that the vector-cjcceleration of a point's 
motion, d^p 

is the vector-velocity in the hodograph. Thus the fundamental pro- 
perties of the hodograph are proved almost intuitively. 
337.] Changing the independent variable, we have 
dp ds , 

if we employ the dash, as before, to denote -5- • 

This merely shews, in another form, that p expresses the velocity 
in magnitude and direction. But a second differentiation gives 

p = vp' + v^p". 
This shews that the vector-acceleration can be resolved into two 
components, the first, vp', being in the direction of motion and 

equal in magnitude to the acceleration of the velocity, t; or -=- ; 


the second, v^p", being in the direction of tha radius of absolute 


196 QUATERNIONS. [338. 

curvature, and having for its amount the square of the velocity 
multiplied by the curvature. 

[It is scarcely conceivable that this important fundamental pro- 
position, of which no simple analytical proof seems to have been 
obtained by Cartesian methods, can be proved more elegantly than 
by the process just given.] 

338.] If the motion be in a plane curve, we may write the 
equation as follows, so as to introduce the usual polar coordinates, 
r and 6, zf 

p = ra"^, 

where a is a unit-vector perpendicular to, ^ a unit-vector in, the 
plane of the curve. 

Here, of course, r and may be considered as connected by one 
scalar equation ; or better, each may be looked on as a function of i. 
By differentiation we get 

29 29 

p = ra^'^ + rdaa'^ ^, 
which shews at once that r is the velocity along, rd that perpen- 
dicular to, the radius vector. Again, 

2£ 29 

which gives, by inspection, the components of acceleration along, 
and perpendicular to, the radius vector. 

339.] For uniform acceleration in a constant direction, we have at 
once, • p = a. 

Whence p = ai + l3, 

where ^ is the vector-velocity at epoch. This shews that the 
hodograph is a straight line described uniformly. 

Also p = —-+fit, 

no constant being added if the origin be assumed to be the position 
of the moving point at epoch. 

Since the resolved parts of p, parallel to /3 and a, vary respect- 
ively as the first and second powers of i, the curve is evidently a 
parabola (§31 (/)). 

But we may easily deduce from the equation the following result, 

T(p + iPa-^^) =-SUa(p + ^ a-^) , 

the equation of a paraboloid of revolution, whose axis is a. Also 

S.a^p = 0, 

34I-J xmEMATics. 197 

and therefore the distance of any point in the path from the point 
— ^/3a~i/3 is equal to its distance from the line whose equation is 

Thus we recognise the focus and directrix property. 

340.] That the moving point may reach a point y we must 
Have, for some real value of t. 

Now suppose Ty3, the velocity of projection, to be given =v, and, 
for shortness, write ot for Uj3. 

Then y = ^i^+viT^. 

Since Tzr = 1, 

we have («2 _ Say) i^ + Ty'^ = 0, 

The values of t'^ are real if 

is positive. Now, as TaTy is never less than Say, it is evident that 
v^ — Say must always be positive if the roots are possible. Hence, 
when they are possible, both values of i^ are positive. Thus we 
hscfefoiir values of t which satisfy the conditions, and it is easy to 
see that since, disregarding the signs, they are equal two and two, 
each pair refer to the same path, but described in ojaposite directions 
between the origin and the extremity of y. There are therefore, if 
any, in general two parabolas which satisfy the conditions. The 
directions of projection are (of course) given by the corresponding 
values of ct. 

341.] The envelop of all the trajectories possible with a given 
velocity, evidently corresponds to 

{v^-Sayf-Ta''Ty^ = Q, 
for then y is the vector of intersection of two indefinitely close paths 
in the same vertical plane. 

Now v^ - Say = TaTy 

is evidently the equation of a paraboloid of revolution of which the 
origin is the focus, the axis parallel to a, and the directrix plane at 

a distance ^r- • 

All the ordinary problems connected with parabolic motion are 

easily solved by means of the above formulae. Some, however, are 

even more easily treated by assuming a horizontal unit-vector in 

198 • QUATERNIONS. [342. 

the plane of motion^ and expressing y3 in terms of it and a. But 
this must be left to the student. 

342.] For acceleration directed to or from a fixed jaoint, we have, 
taking that point as origin, and putting P for the magnitude of 
the central acceleration, 

P =PUp. 

Whence, at once, f^pp = 0. 

Integrating, Fpp = y = a constant vector. 

The interpretation of this simple formula is — first, p and p are in 
a plane perpendicular to y, hence the path is in a plane (of course 
passing through the origin) ; second, the area of the triangle, two 
of whose sides are p and p is constant. 

[It is scarcely possible to imagine that a more simple proof than 
this can be given of the fundamental facts, that a central orbit is a 
plane curve, and that equal areas are described by the radius vector 
in equal times.J 

343.] When the law of acceleration to or from the origin is that of 
the inverse square of the distance, we have 

p_ M 


where p. is negative if the acceleration be directed to the origin. 
Hence p = ^ . 

The following beautiful method of integration is due to Hamilton. 

(See Chapter IV.) 

dJJp Vp.Vpp Up.y 

Generally, ^^ = - -^^ =--f^' 

, n .. dUp 

therefore py = —p. —j- , 

and py = e—pJJp, 

where e is a constant vector, perpendicular to y, because 

Sy'p = 0. 
Hence, in this case, we have for the hodograph, 
p = iy"^ — fji,Up.y~\ 
Of the two parts of this expression, which are both vectors, the 
first is constant, and the second is constant in length. Hence the 
locus of the extremity of p is a circle in a plane perpendicular to y 

(i.e. parallel to the plane of the orbit), whose radius is ^ > and 
whose centre is at the extremity of the vector ey""^. 

[This equation contains the whole theory of the Circular Hodo- 

345-] KINEMATICS. 199 

graph. Its consequences are developed at length in Hamilton's 

344. J We may write the equations of this circle in the form 

y(p-ey-^) = Yy' 

(a sphere), and /Syp = 

(a plane through the origin, and through the centre of the sphere). 
The equation of the orbit is found by operating by Y.p upon that 
of the hodograph. We thus obtain 

y = r.pey-i + ^y/Dy-i, 
or y2 =Sip + ix.Tp, 

or txTp = Se{y^e-'^-p)-, 

in which last form we at once recognise the focus and directrix 
property. This is in fact the equation of a conicoid of revolution 
about its principal axis (e), and the origin is one of the foci. The 
orbit is found by combining it with the equation of its plane, 

Syp = 0. 
We see at once that y^ e^^ is the vector distance of the directrix 

. . . Te 
from the focus ; and similarly that the eccentricity is — j and the 

. -2My^ '' 

maior axis — = =- • 

345.] To take a simpler case : let the acceleration vary as the dis- 
tance from the origin. 

Then p = ±m^p, 

the upper or lower sign being used according as the acceleration is 
from or to the centre. 

This is (^ + «.2)p = 0. 

Hence p = ae'"«+i3£-™'i 

or p = a cos mt + fi sin mt, 

where a and j3 are arbitrary, but constant, vectors; and e is the 

base of Napier's logarithms. 

The first is the equation of a hyperbola (§ 31, ^) of which a and ft 
are the directions of the asymptotes ; the second, that of an ellipse 
of which a and ft are semi-conjugate diameters. 

Since p == m {as'^ — fts'""} , 

or = m {—a sin mt + ft cos mt}, 

the hodograph is again a hyperbola or ellipse. But in the first 
case it is, if we neglect the change of dimensions indicated by the 

200 QUATERNIOKS. [346. 

scalar factor m, conjugate to the orbit ; in the case of the ellipse it 
is similar and similarly situated. 

346.] Again, let the acceleration he as the inverse third power of 
the distance, we have aUp 

Of course, we have, as usual, 

Vpp = y. 
Also, operating by S.p, 

... (xSpp 

of which the integral is u 

the equation of energy. 

Again, Spp = -^ ■ 

Hence Spp + p'^ = C, 

or Spp = Ct, 

no constant being added if we reckon the time from the passage 
through the apse, where Spp = 0. 

We have, therefore, by a second integration, 

p^ = Cfi + C' (1) 

[To determine C", remark that 
pp = Ct + y, 
or pV = CH^-y'^. 

But p^p^ = Cp^—jj. (by the equation of energy), 

= CH^ + CC'-^, by(l). 
Hence CC'= ju-y^.] 

To complete the solution, we have, by § 133, 

where /3 is a unit-vector in the plane of the orbit. 

But r^ = - „ 

p p^ 

t — _y_. 


Hence ^°^~i" ^ "V i 

^ ~ '^JCt^ + C 

The elimination of t between this equation and (1) gives Tp in 

terms of Up, or the required equation of the path. 

We may remark that if d be the ordinary polar angle in the 

orbit, T/o 

log^ = eUy. 

348-] KINEMATICS, 201 

Hence we have = —Ty I - 

W + G' \ 
and r^=-{Gt''-\-C'\ ) 

from which the ordinary ec[uations of Cotes' spirals can be at once 
found. [See Tait and Steele's Dynamics of a Particle, third edition, 
Appendix (A).] 

347.] To find the conditions that a given curve may he the hodo- 
graph corresponding to a central orbit. 

If or be its vector, given as a function of the time, f^ndt is that of 
the orbit ; hence the requisite conditions are given by 

Tvjftsdt ■=■ y, 
where y is a constant vector. 

We may transform this into other shapes more resembling the 
Cartesian ones. 

Thus ^ FijfijTdt = 0, 

and * VzrfiiTdt+Vm'ST = 0. 

From the first f'^dt = x-ir, 

and therefore xYTsis = jf, 

or the curve \& plane. And 

m T^is + VisTs = ; 
or eliminating x, yViim = —(Fm-wY- 

Now if v' be the velocity in the hodographj 2if its radius of curva- 
ture, p' the perpendicular on the tangent ; this equation gives at 
once hv'= R'p"^, 

which agrees with known results. 

348.] The equation of an epitrochoid or hypotrochoid, referred to 
the centre of the fixed circle, is evidently 

p = ai " a + M " a, 
where a is a unit-vector in the plane of the curve and i another 
perpendicular to it. Here o> and co^ are the angular velocities in 
the two circles, and t is the time elapsed since the tracing point 
and the centres of the two circles were in one straight line. 
Hence, for the length of an arc of such a curve, 

s —fT'pdt =fdt V { (o^a 2 + 2 tocoi a* cos (o) - (Oi) i! + ffli^ «2 } ^ 

= I dt ^y \{Q)a + <i)-J)f ±'i:u>a>^ah . ^ — o""^ 3' 
which is, of course, an elliptic function. 

202 QUATERNIONS. [349- 

But when the curve becomes an epicycloid or a hypocycloid, 
coa+w^J = 0, and 

which can be expressed in finite terms, as was first shewn by Newton 
in the Principia. 

The hodograph is another curve of the same class, whose equa- 
tion is 2iat 2a)j< 

p = i{aa)i " a + bco-yi ^ a); 
and the acceleration is denoted in magnitude and direction by the 
vector iat 2M\t 

p = —au?i " a—ba\ i " a. 

Of course the equations of the common Cycloid and Trochoid may 
be easily deduced from these forms by making a indefinitely great 
and o) indefinitely small, but the product aa> finite ; and transferring 
the origin to the point p=. aa. 

349.] Let i be the normal-vector to any plane. 
Let la- and p be the vectors of any two points in a rigid plate in 
contact with the plane. • 

After any small displacement of the rigid plate in its plane, let 
dm and dp be the increments of m and p. 

Then Sidm = 0, Sidp = ; and, since T^sr — p) is constant, 
S{-a-—p) idsr—dp) = 0. 
And we may evidently assume 

dp ■= a)i{p — t), 
dsT =: Q>J(or — t) ; 
where of course t is the vector of some point in the plane, to a rota- 
tion 0) about which the displacement is therefore equivalent. 
Eliminating it, we have 

d('ST — p) 

m = -^ —) 

•ST — p 

which gives to, and thence r is at once found. 
For any other point a- in the plane figure 
Sida- = 0, 
S{p—(t) {dp— da) = 0. Hence dp— da = aji«(p— o-). 
S{(T—'m){dijr — d(T) = 0. Hence dcr—dzr = oo^i^a—w). 
From which, at once, coj = w^ = co, and 
da- = (0? ((T— t), 
or this point also is displaced by a rotation a> about an axis through 
the extremity of r and parallel to i. 

35 1 -J KINEMATICS. 203 

350.] In the ease of a rigid body moving about a fixed point 

let OT, p, a- denote the Ysctors of any three points of the body ; the 

fixed point being origin. 

Then ■sr', p^, <j^ are constant^ and so are Sssp, Spar, and Sa-sr. 

After any small displacement we have, for tn- and p, 

Smdzs- = 0, ^ 

Spdp = 0, i (1) 

Szjdp + Spdzj = 0. ) 

Now these three equations are satisfied by 

«?sr =: VazT, dp = V^ap, 

where a is any vector whatever. But if dur and dp are given, then 

Vdsrdp ^ T^.FazrFap = aS.ap^. 

Operate by S.V^p, and remember (1), 

S^zrdp = S^pd^ = S^.apTn: 

Vdvrdp Vdpdxs ,. 

^^^'^^ «= -s^ = -s^ ' ^') 

Now consider o-, Strdv = 0, \ 

Spdcr = —Strdp, V 
Sssda = — Sa-duT. ) 
da = Va<T satisfies them all, by (2), and we have thus the proposi- 
tion that ani/ small displacement of a rigid body abont a ^xed point is 
equivalent to a rotation. 

351.1 To represent the rotation of a rigid body about a given awis^ 
ihrougli a given finite angle. 

Let a be a unit-vector in the direction of the axis, p the vector 
of any point in the body with reference to a fixed point in the axis, 
and 6 the angle of rotation. 

Then p = a''^Sap + a-^Vap, 

=■ — aSap — a Vap. 

The rotation leaves, of course, the first part unaffected, but the 
second evidently becomes 

— a ^ aVap, 
or — a Vap cos 6 + Vap sin 6. 

Hence p becomes 

pj z= — aSap — a Vap COS -f- Vap sin d, 

= (cos- + asm-jp(cos--asm-j. 
= a pa 

204 QUATERNIONS. [352. 

352.] Hence to compound two rotations about axes which meet, we 
may evidently write, as the effect of an additional rotation <^ about 
the unit-vector ;8, ^ _* 

Hence p^ = P' a" pa~ " p~' . 

If the /3-rotation had been first, and then the a-rotation, we should 

have had 1 ± _* _i. 

and the non-commutative property of quaternion multiplication 
shews that we have not, in general, 

P'i = ft- 
If a, fi, y be radii of the unit sphere to the corners of a spherical 

triangle whose angles are - > ^ . - > we know that 

U &i u ^ 

y" /3 " o "^ = — 1 . (Hamilton, Lectures, p. 267.) 

if. * 

Hence /3'o'=— y"'', 

-t ± 
and we may write P2 = y " py^^ 

or, successive rotations about radii to two corners of a spherical triangle, 

-and through angles double of those of the triangle, are equivalent to a. 

single rotation about the radius to the third corner, and through an 

angle double of the exterior angle of the triangle. 

Thus any number of successive _/?«Jfe rotations may be compounded 
into a single rotation about a definite axis. 

353.] When the rotations are indefinitely small, the effect of 
one is, by § 351, p^ = f,-\-OiVap, 

and for the two, neglecting products of small quantities, 

p^ = p-\-(xTap+W^P, 
a and b representing the angles of rotation about the unit-vectors 
a and ^ respectively. 

But this is equivalent to 

P2 = p + r(aa-hb^)FV(aa + bi3)p, 
representing a rotation through an angle T{fi,a + b^), about the unit- 
vector TJ((xa + 6)3). Now the latter is the direction, and the former 
the length, of the diagonal of the parallelogram whose sides are 
(xa. and b/8. 

We may write these results more simply, by putting a for <ya, 
/3 for b/3, where a and ^ are now no longer unit-vectors, but repre- 

355-1 KINEMATICS. 205 

sent by their versors tlie axes, and by their tensors the angles (small)j 
of rotation. 

Thus pj^ = p+ Vap, 

P2 = p+Fap+ V^p, 
= p+Fia + p)p. 

354.] The general theorem, of which a few preceding sections 
illustrate special cases, is this : 

By a rotation, about the axis of q, through double the angle of q, 
the quaternion r becomes the quaternion qrq~^. Its tensor and 
angle remain unchanged, its plane or axis alone varies. 

A glance at the figure is sufficient for . q 

the proof, if we note that of course 
T.qrq"^^ Tr, and therefore that we need 
consider the versor parts only. Let Q 
be the pole of q, 

A£=q, JJB' = q-\ WC' = r. 

Join C'A, and make AG = C'A. Join 

Then CB is qrq-'^, its arc CB is evidently equal in length to that 
of r, B'C; and its plane (making the same angle with B'B that 
that of B'C does) has evidently been made to revolve about Q, the 
pole of q, through double the angle of q. 

If r be a vector, = p, then qpq"^ (which is also a vector) is the 
result of a rotation through double the angle of q about the axis 
of q. Hence, as Hamilton has expressed it, if B represent a rigid 
system, or assemblage of vectors, 

is its new position after rotating through double the angle of q 
about the axis of q. 

355.] To compound such rotations, we have 
r.qBq'^.r^''- =rq.B.{rq)-^. 

To cause rotation through an angle ^-fold the double of the angle 
of q we write q^Bq-K 

To reverse the direction of this rotation write q~^BqK 

To translate the body B without rotation, each point of it moving 
through the vector a, we write a + B. 

To produce rotation of the translated body about the same axis, 
and through the same angle, as before, 

q{a + B)q-\ 

Had we rotated first, and then translated, we should have had 

a + qBq-'^. 




The obvious discrepance between these last results might perhaps 
be useful to those who do not believe in the Moon's rotation, but 
to such men quaternions are unintelligible. 

356.] Given the instantaneous axis in terms of the time, it is re- 
quired to find the single rotation which will bring the body from any 
initial position to its position at a given time. 

If a be the initial vector of a point of the body, ot the value of 
the same at time t, and q the required quaternion, we have 

^ = i°r^ (1) 

Differentiating with respect to t, this gives 

■ar = qaq~^—qaq''^qq~^, 

= 2r.{rqq-\qaq-^). 
But ■a^ = Vei!7 = V.eqaq~^. 

Hence, as qaq"^ may be any vector whatever in the displaced 

body, we must have e = 2 Tqq-^ (2) 

This result may be stated in even a simpler form than (2), for we 
have always, whatever quaternion q may be, 


Vqq-^ = 



and, therefore, if we suppose the tensor of q, which may have any 
value whatever, to be a constant (unity, for instance), we may write 

(2) in the form eq = 2q (3) 

An immediate consequence, which will be of use to us later, is 

q.q-''eq = 2q (4) 

357.J To express q in terms of the usual angles i/f, 6, ^. 
Here the vectors i, J, h in the original position of the body corre- 

•spond to OA, OB, 00, respectively, 
at time t. The transposition is ef- 
fected by — first, a rotation y]r about 
k ; second, a rotation 6 about the 
new position of the line originally 
coinciding with/; third, a rotation 
(^ about the final position of the line 
at first coinciding with k. 

Let i, j, k be taken as the initial 
directions of the three vectors which 
at time t terminate at A, B, C re- 
The rotation >/f about h has the operator 

t _i 
k''{ )k ''. 

357-] KINEMATICS. 207 

This converts y into r), where 

t -i- . 
'tj = k''ji " = J COS yp—i sin \{r. 

The body next rotates about tj through an angle 9. This has 

the operator t _* 

It converts k into 

^ * _^ 6 Q Q 6 

OC = C= yfk-q "= (cos- +'?sin-)/4(cos-— jjsin-) 

= ^cos04sind(jcos\/f + ysin\/f). 
The body now turns through the angle (p about C the operator 
being * _* 


= (eos - + C sin -) (cos - + j? sin -) (cos| + k sin |) 

= [cos~ + Csm-) cos-cos^ + Acos-sin|^ 

. -Jf , . . . , . 6 . if ,. . . ~[ 

+ sin-cos^(_;cosT//— »sini|f) + sin-sin^(«eosi/f+_;smi/f) 

a 2 2 2 _J 

/ <^..^sr e ^// . . e . f . . e ^ir , ■ ■^i 

= (cos-+Csin-) cos-cos^ — »sm-sin-i^ + » sin -cos- + /ecos-sin— 
\ 2 ^ 2''L2 2 2 2-' 2 2 2 2j 

4> ^ . (b . 6 . ■Jf . „ 
= cos — cos -cos— + sm — sin-sin— sm^cosiir 
222222 ^ 

. A . e ^ ■ „ . , .</> e . ■jf 

— sm— sm-eos — sin5sm\/f— sm — cos- sin — cos ^ 
2 2 2 ^ 2 2 2 

.^ (j) . 6 . -f . <j) 6 ^ . „ 

+ «( — cos — sin - sin — + sm — cos - cos — sm 6 cos <|f 
V 2 222 2'2 ^ 

. d) . 6 -Jf „ . (b e . \1/ . „ . N 

— sm — sm - cos ^ cos 9 + sm — cos - sin — sm sin i/j ) 

• 2 2 2 2 2 2 ^■' 

.f (b . d lif . (b e ^ . „ . 
+ f I cos — sm - eos — + sm — cos - eos — sm a sm \|f 
■^^222222 ^ 

. <b . e . f „ . <f) . -^ . „ ,\ 

— sm — sin - sm j- cos 0— sin — cos - sin sm 6 eos i/f ) 

2 2 2 2 2 2 ^>' 

7/0 . yj/ . <l) if a 
+ ^ I cos — cos - sin — + sm ^ cos - cos — cos 9 
V2 2 '2 2 2 2 

.rf).e.>|f.^., . (h . ir . „ x 

+ sm5-sin-sin — sm0sinvf+ sm— sin- cos— sm9cos\/r) 

222 ^222 ^/ 

rf) + i/f 5 . . (b—yif . . (b—yJf . , . <b + ^lt 6 
= cos cos - + « sm sm - +_; cos sin- + a sm -^^ — 2: cos- > 

which is, of course, essentially unsyuimetrical. 

208 QUATERNIONS. [358. 

358.] To find the usual equations connecting \j/, 6, (p with the an- 
gular velocities about three rectangular axes fixed in the body. 

Having tlie value of q in last section in terms of the three angles, 
it may be useful to employ it, in conjunction with equation (3) of 
§ 356, partly as a verification of that equation. Of course, this is 
an exceedingly roundabout process, and does not in the least re- 
semble the simple one which is immediately suggested by qua- 

We have 2q=. eq= {<a^OJ + oo20£ + <agOC} q, 

■whence ^i~^4 = S~^ {o)iOA + ai^ OB + co^ 00} q, 
or 2q = q{ia)i +ju>2 + ka^). 

This breaks up into the four (equivalent to three independent) 

2 -7; ( cos ^^-—-!- cos-) 
dt^ 2 2> 

. d)—-J/ . 6 d> — -Jf . e . (b + \lf e 

= — CO, sm — — -^ sin - — o), cos - — — sin co„ sm - — — cos - j 

^2 2^2 2^2 2 

2-T:(sin ^„ ^ sin-) 
dt V 2 2'' 

(t> + \l/ e . (b + xir 6 (h — ylr . 

= 0), cos — - — cos- — coosm — — ^eos- + a),cos-^^ — i-sm-, 
1 2 2 ^ 2 2 ^ 2 2 

^d ^ (b — \lr . 0\ 

2 — (cos „ sin-l 

dt\ 2 2' 

. (t> + ^ (b + ylf . tb — yS, . 

= Q)j sin cos - + i»2 cos cos ;r — 0)3 Sin — — ~ sm - J 

2 ^ ( sin ^ cos - ) 
dt^ 2 2' 

(b—-d/ . . <b — ^ . (h + y], 

= —6)1 cos sin- + 6)2 sin — —^ sm- + a>3C0S^!^ — 31 cos-- 

Prom the second and third eliminate 0— x/^, and we get by in- 
spection ^ a / ■ i , ,^ ^ 
^ COS - . = (uj sm <p + 0)2 cos (^) cos - > 

or ^ = Wj sin (/) + Wg cos (/). (1) 

Similarly, by eliminating between the same two equations, 

. 0,- -.s ■ ^ ^ ^ • ^ 

sm — ((^ — \\r) = cog sin — + iBi cos 9 cos — — a>^ sm <p cos— • 
2 i £1 2 

And from the first and last of the group of four 

■ ■ . . . 

cos-{(l> + \j/) = 0)3 cos- — WjCosc^ sm- + Wg sm <^ sm - • 
2 2 i ^ 

359-] KINEMATICS. 209 

These last two equations give 

<j) + \jfCOsd = 0)3 (2) 

<p cos 6 + \jf = ( — w^ cos (^ + 0)2 sin (^) sin + 0)3 cos0. 
From the last two we have 

■\jr sin 6 =— ctfj 008(^ + 0)2 sin (^ (3) 

(1), (2), (3) are the forms in which the equations are usually given. 

359.] To deduce eiepressions for the direciion-coaines of a set 0/ 
rectangular axes in any position in terms of rational functions of three 

Let a, yS, y be unit- vectors in the directions of these axes. Let c[ 
be, as in § 356, the requisite quaternion operator for turning the 
coordinate axes into the position of this rectangular system. Then 

q^ = w + xi-^yj-^zh, 
where, as in § 356, we may write 

1 = W^+iB^+y^+^2. 

Then we have (f^ =■ w—xi—yj-\-zk, 

and therefore 

a = qiq"^ = {wi—x—yk + zf){w—xi—yj—z}c) 

= [ie^ +x^ —y'^ —z^)i+ 2 {wz + xy)J +2(xz—toy)k, 
where the coefficients of i, J, k are the direction-eosines of a as 
required. A similar process gives by inspection those of ^ and y. 

As given by Cayley*, after Rodrigues, they have a slightly 

different and somewhat less simple form — to which, however, they 

are easily reduced by putting 

_'^_.5'_'^_ ^ 
\ jJ. V ^i 

The geometrical interpretation of either set is obvious from the 

nature of quaternions. For (taking Cayley's notation) if be the 

angle of rotation : cos^ cosy, cos h, the direction-cosines of the axis, 

we have 

6 6 

q = w + xi+yj + zJc = cos- + sin- (i cos/ +/ cosy + /i cos ^), 


SO that w — cos - > 

X = sm-cos/, 


y = sm-cosy; 

• ^ i 
z = sm - cos n. 

* Camb. and Bub. Math. Journal. Vol. i. (1846.) 

210 QUATERNIONS. [360. 

From these we pass at once to Rodrigues' subsidiary formulae, 

K = -5 = sec^ - . 
w^ 2 

X = — = tan - cos/, 

&c. = &c. 

360.J By the definition of Homogeneous Strain, it is evident that 
if we take any three (non-eoplanar) unit-vectors a, /3, y in an un- 
strained mass, they become after the strain other vectors, not neces- 
sarily unit- vectors, a^, ySj^, y^. 

Hence any other given vector, which of course may be thus ex- 
pressed, p =i xa + yfi -\- zy, 
becomes Pi = c(ia-^-^y^^->r zy-^, 
and is therefore known if a^, j3^, yj be given. 
More precisely 

pS.afiy = aS.j3yp + j3S.yap + yS.al3p 

piS.a^y = (ppS.a^y = aj^S.^yp + ^-j^S.yap + yiS.a^p. 

Thus the properties of cf), as in Chapter V, enable us to study with 
great simplicity strains or displacements in a solid or liquid. 

For instance^ to find a vector whose direction is unchanged hy the 
strain, is to solve the equation 

Yp^p = 0, or <^p = gp, 
where ^ is a scalar unknown. 

[This vector equation is equivalent to three simple equations, and 
contains only three unknown quantities ; viz. two for the direction 
of p (the tensor does not enter, or, rather, is a factor of each side), 
and the unknown ^.] 

We have seen that every such equation leads to a cubic in g 
which may be written 

g^—m^g'^ + m^g—m = 0, 
where ««2i ^u ''"■ ^'"^ scalars depending in a known manner on the 
constant vectors involved in 0. This must have one real root, and 
may have three. 

361. J For simplicity let us assume that a, /3, y form a rectangular 
system, then we may operate by S.a, S.^, and S.y; and thus at 
once obtain the equation for g, in the form 

0... (1) 

Saoj^ 4 g. 












To reduce this we have 

'S'aoi, Sa^i, Say^^ 
S^a^, Sj3p^, Sl3y^ 
Sya^, Sy^i, Syy^ 

1 S^aa^ + S^pa^ + S^ya,, ^Saa^SajS^, 

Sya^, Syl3i, 

which, if the mass be rigid, becomes successively 


Saa^ \Sypj^, 


= s^mMyri-7iSyPi) 

Thus the equation becomes 

- 1 -ff{Saa^ + aS^^i + Syy^) +g^ {Saa^ + Sp^^ + Syy^) +g^ = 0, 
{g-^){9^+9{^+Saa^ + Spp^ + 8yy;)+l) =0. 

362.] If we take Tp :=G we consider a portion of the mass 
initially spherical. This becomes of course 




an ellipsoid, in the strained state of the body. 

Or if we consider a portion which is spherical after the strain, i. e 
Tp^ = C, 
its initial form was T^p = C, 

another ellipsoid. The relation between these ellipsoids is obvious 
from their equations. (See § 311.) 

In either case the axes of the ellipsoid correspond to a rectangular 
set of three diameters of the sphere (§ 257). But we must care- 
fully separate the cases in which these corresponding lines in the 
two surfaces are, and are not, coincident. For, in the former case 
there is jmre strain, in the latter the strain is accompanied by ro- 
tation. Here w6 have at once the distinction pointed out by 
Stokes* and Helmholtzf between the cases of fluid motion in 
which there is, or is not, a velocity-potential. In ordinary fluid 
motion the distortion is of the nature of a pure strain, i.e. is differ- 
entially non-rotational ; while in vortex motion it is essentially ac- 
companied by rotation. But the resultant of two pure strains is 
generally a strain accompanied by rotation. The question before us 
beautifully illustrates the properties of the linear and vector function. 

* Cambridge Phil Trans. 1845. 

+ Crelle, vol. Iv. 1857. See also Phil Mag. (Supplement) June 1867. 

P 2 

212 QUATEENIONS. [363. 

363.] To find the criterion of a pure strain. Take a, p, y now as 
unit-vectors parallel to the axes of the strain-ellipsoid, they become 
after the strain a a, bj3, cy. 

Hence p, = (pp ——aaSap—b^S^p — cySyp. 

And we have, for the criterion of a pure strain, the property of the 
function <\>, that it is self-conjugate, i. e. 

Sp(fi<T = S(T<pp. 

364.J Two pure strains, in succession, generally give a strain ac- 
companied hy rotation. For if <p, \jf represent the strains, since they 
are pure we have Sp^a- = Sai^tp, ^ 

But for the compound strain we have 

Pi = XP = ^^P> 
and we have not generally 

Spx<T = Saxp. 

For 8p^<ji<T = Sa-(j)\jfp, 

by (1), and i/?0 is not generally the same as (f)\j/. (See Ex. 7 to 
Chapter V.) 

365.] The simplicity of this view of the question leads us to 
suppose that we may easily separate the purs strain from the rotation 
in any case, and exhibit the corresponding functions. 

When the linear and vector function expressing a strain is self- 
conjugate the strain is pure. When not self-conjugate, it may be 
broken up into pure and rotational parts in various ways (analogous 
to the separation of a quaternion into the sum of a scalar and a 
vector part, or into the product of a tensor and a versor part), of 
which two are particularly noticeable. Denoting by a bar a self- 
conjugate function, we have thus either 

€l> = qw{ )q-\ or 4, = ^^.q{ )q-\ 

where e is a vector, and q a quaternion (which 'may obviously be 
regarded as a mere versor). 

That this is possible is seen from the fact that (j) invofves nine 
independent constants, while ^ and ct each involve six, and e and q 
each three. If </>' be the function conjugate to <f>, we have 

<j>'=f-F.e{ ), 
60 that 2\}r = <p + <f)', 

and 2r.i{ ) = 0-()i', 

which completely determine the first decomposition. This is, of 

365.] KINEMATICS. 213 

course, perfectly well known in quaternions, but it does not seem 
to have been noticed as a theorem in the kinematics of strains that 
there is always one, and but one, mode of resolving a strain into the 
geometrical composition of the separate effects of (1) a pure strain, 
and (2) a rotation accompanied by uniform dilatation perpendicular 
to its axis, the dilatation being measured by (sec. 0—\) where Q is 
the angle of rotation. 

In the second form (whose solution does not appear to have been 
attempted), we have 

<t> = i^{ ) r\ 

where the pure strain precedes the rotation, and from this 

0'=:^.j-i( )q, 
or in the conjugate strain the rotation (reversed) is followed by the 
pure strain. From these 

and OT is to be found by the solution of a biquadratic equation*. 
It is evident, indeed, from the identical equation 

S.CT<p'(l)p = S.p(j/(I><T 

that the operator ^'^ is self-conjugate. 
In the same way 

<^<^'( )=q^^{s-H )q)q-\ 
or §■-! {4,<t)'p) q = ^^ iq-'^pq) = ¥^ [q'^pq), 

which shew the relations between ^<^', <^'0, and q. 
To determine q we have 

<t>p-q = q^P 

* Suppose the cubic in ct to be 

ra^' + 3^"" + gr, TO^ + 32 = 0, 
write 0; for ^'<j> in the given equation, and by its help this may be written as 

(W + sf)a) + jriW= + g'2 = = w'(o) + g',)+araj + 5f2. 
Eliminating 5=, we have 

<"' + (2?, -ff") oi' + {g,''-2gg,)o,-g^ = 0. 
This must agree with the (known) cubic in ai, 

0^ -i- mar' + m^a + ma=0, 
suppose, so that by comparison of coefficients we have 

so that g, is known, and g= ' • 

2 -v/— ma 

where 2^. = m-(^^^ 

The values of the quantities g being found, w is given in terms of <u by the equation 
above. (Proc. B. S. E., 1870-71-) 

214 QUATERNIONS. [366. 

whatever be p, so that 

S.Fq{<f)—m)p= 0, 

or S.p{<^'-^)Fq = Q, 

which gives {'¥~ ^) ^i = 0> 

The former equation gives evidently 

whatever be o and /3 ; and the rest of the solution follows at once. 
A similar process gives us the solution when the rotation precedes 
the pure strain. 

366.] In general, if 

Pi = (jyp = —CiSap—^j^S^p—y-^Syp, 
the angle between any two lines, say p and a; becomes in the 
altered state of the body 

cos-^ {-S.U<l)pU<l><T). 
The plane *Sfp = becomes (witji the notation of § 144) 

SCpi = = SC<l>p = Sp<l>'C 
Hence the angle between the planes SCp = 0, and Srjp = 0, which 
is cos~^(—iS.UCUri), becomes 

The locus of lines equally elongated is, of course, 
T^Up = e, 
or T<i,p = eTp, 

a cone of the second order. 

367.] In the case of a Simple Shear, we have, obviously, 
Pi = <i>P = p + fiSap, 
where Sa^ =0. 

The vectors which are unaltered in length are given by 

Tp^ = Tp, 
or 2 S^pSap + l3^S^ap = 0, 

which breaks up into S. ap = 0, 

and Sp{2fi + fi^a) = 0. 

The intersection of this plane with the plane of a, /3 is perpen- 
dicular to 2/3 + /3*a. Let it be a + a? /3, then 
-S.(2/3 + y32a)(a + a;/3) = 0, 
i.e. 2a! — 1 = 0. 

Hence the intersection required is 

368.] KINEMATICS. 215 

For the axes of the strain, one is of course aj3, and the others 

are found by making TcjyJJp a maximum and minimum. 

Let p = a + x^, 

then pi= (pp = a + xj3—l3, 

and -^ = max. or mm., 



gives x^—x+-^ = 0, 

from which the values of x are found. 
Also, as a verification, 

S.{a + XiP){a + X2l3) =—l + p.'^x^x^, 
and should be = 0. It is so, since, by the equation, 

_ 1 


S{a + {x^-l)fi} {a-\-{x^-\)p} =-\+&^{x^x^-{x^ + x.,)+l}, 
which ought also to be zero. And, in fact, aj^ + ^g = 1 by the equa- 
tion ; so that this also is verified. 

368.] We regret that our limits do not allow us to enter farther 
upon this very beautiful application. 

But it may be interesting here, especially for the consideration 
of any continuous displacements of the particles of a mass, to in- 
troduce another of the extraordinary instruments of analysis which 
Hamilton has invented. Part of what is now to be given has been 
anticipated in last Chapter, but for continuity we commence afresh. 

If Fp = C (1) 

be the equation of one of a system of surfaces, and if the differential 

of (l)be Svdp= 0, (2) 

v is a vector perpendicular to the surface, and its length is inversely 
proportional to the normal distance hetween two consecutive surfaces. 
In fact (2) shews that v is perpendicular to dp, which is any tangent 
vector, thus proving the first assertion. Also, since in passing to a 
proximate surface we may write 

Svbp = 8C, 
we see that F{p + v-^hC) = C + W. 

This proves the latter assertion. 

It is evident from the above that if (1) be an equipotential, or an 
isothermal, surface, —v represents in direction and magnitude the force 
at any point or the flux of heat. And we have seen (§ 317) that if 

. d . d -. d 
dx '' dy dz 

216 QUATERNIONS. [369. 

d'' A^ d^ 
gmng v^=______, 

then V = VFp. 

This is due to Hamilton (Lectures on Quaternions, p. 611). 

369.] From this it follows that the effect of the vector operation 
V, upon any scalar function of the vector of a point, is to produce 
the vector which represents in magnitude and direction the most rapid 
change in the value of the function. 

Let us next consider the effect of V upon a vector function as 

<^ = ii+Jv + ^C- 

We have at once 


and in this" semi-Cartesian form it is easy to see that : — 

If T represent a small vector displacement of a point situated at 

the extremity of the vector p (drawn from the origin) 

SV a- represents the consequent cubical compression of the group 

of points in the vicinity of that considered, and 

VVa represents twice the vector axis of rotation of the same 

group of points. 

Similarly 5. V= - (^^ +, i- + C^) = -D., 

or is equivalent to total differentiation in virtue of our having 
passed from one end to the other of the vector a. 

370.] Suppose we fix our attention upon a group of points which, 
originally filled a small sphere about the extremity of p as centre, 
whose equation referred to that point is 

To3 = e (1) 

After displacement p becomes p + a-, and, by last section, p + a> 
becomes p + m + cr— (jSa)V)o-. Hence the vector of the new surface 
which encloses the group of points (drawn from the extremity of 

p + tr) is Q)i = oi — {8<i>V)(T (2) 

Hence o) is a homogeneous linear and vector function of w-^ ; or 

and therefore, ^7 (1)> ^^o)i = e, 

the equation of the new surface, which is evidently a central surface 

of the second order, and therefore, of course, an ellipsoid. 

We may solve (2) vsdth great ease by approximation, if we re- 
member that T^ is very small, and therefore that in the small term 
we may put <Bj for w ; i. e. omit squares of small quantities ; thus 
(o = <Bj + (Sa>jV)a: 

372-] KINEMATICS. '217 

371.] If the small- displacement of each point of a medium is in the 
direction of, and proportional to, the attraction exerted at that point 
hy any system of material masses, the displacement is effected without 

For \i Fp = C be the potential surface^ we have Sddp a complete 
differentia] ; i. e. in Cartesian coordinates 
^dx + r]di/ + (dz 
is a differential of three independent variables. Hence the vector 
axis of rotation ^ ^^ g 

vanishes by the vanishing of each of its constituents, or 

r.Va- = 0. 

Conversely, if there he no rotation, the displacements are in the 
direction of and proportional to, the normal vectors to a series of 

For 0=r.dpr.Vcr = (SdpV) a- - ^Sadp, 

where, in the last term, V acts on o- alone. 

Now, of the two terms on the right, the first is a complete differ- 
ential, since it may be written —Dcip(T, and therefore the remaining 
term must be so. 

Thus, in a distorted system, there is no compression if 

SVa- = 0, 
and no rotation if V.Va = ; 

and evidently merely transference if o- = a = a constant vector, 
which is one case of Vg- = q. 

In the important case of a- = eVFp 
there is evidently no rotation^ since . 

Vff = eV^Fp 
is evidently a scalar. In this case, then, there are only translation 
and compression, and the latter is at each point proportional to the 
density of a distribution of matter, which would give the potential 
Fp. For if r be such density, we have at once 
V^Fp = 4 7rr*. 
372.] The Moment of Inertia of a body about a unit vector a as 
axis is evidently jfp = -■2m{rapf, 

where p is the vector of the portion m of the mass, and the origin 
of p is in the axis. 

« Proc. B. 8. K, 1862-3, 

218 QUATERNIONS. [372. 

Hence if we take hTa = e^, we have, as locus of the extremity of a, 
Jfe* =—^m,{Japf = MSai^a (suppose), 
the momental ellipsoid. 

If ts be the vector of the centre of inertia, o- the vector of m with 
respect to it, we have p = ot + o- ; 

therefore MB =-^m{{ Va^f + ( Faaf } 

= -M{ Vamf + MSa<i,^a. 
Now, for principal axes, Jc is max., min., or max.-min., with the 
condition ^z = _ 1 . 

Thus we have Sa{'arFaz7—(f)ia) — 0, 

Saa = ; 
therefore — ^la + wFatiT = ^a = h^o. (by operating by So). 

Hence (<^-^-\-k'^-\-vs^)a = +cr<S'aOT (1) 

detei-mines the values of a, Ic^ being found from the equation 

<St!r(<^ + P + OT2)-lt!7 = 1 (2) 

Now the normal to AS(r(0 + P + OT2)-^(7 = 1, (3) 

at the point o- is ((/> + ^^ + ot^)"^ o-. 

But (3) passes through — sr, by (2), and there the normal is 

which, by (1), is parallel to one of the required values of a. Thus 
we prove Binet's theorem that the' principal axes at any point are 
normals to the three surfaces, eonfocal with the momental ellipsoid, 
which pass through that point. 


1. Form, from kinematical principles, the equation of the cycloid ; 
and employ it to prove the well-known elementary properties of the 
arc, tangent, radius of curvature, and evolute, of the curve. 

2. Interpret, kinematically, the equation 

p = aU{pt-p), 
where /3 is a given vector, and a a given scalar. 

Shew that it represents a plane curve ; and give it in an in- 
tegrated form independent of t. 


3. If we write ct = ^i—p, 
the equation in (2) becomes 

/3 — ■nr = aUv7. 
Interpret this kinematically ; and find an integal. 

What is the nature of the step we have taken in transforming 
from the equation of (2) to that of the present question ? 

4. The motion of a point in a plane being given, refer it to 

{a.) Fixed rectangular vectors in the plane. 

{b.) Rectangular vectors in the plane, revolving uniformly 
about a fixed point. 

(c.) Vectors, in the plane, revolving with different, but uni- 
form, angular velocities. 

{d.) The vector radius of a fixed circle, drawn to the point of 

contact of a tangent from the moving point. 
In each case translate the result into Cartesian coordinates. 

5. Any point of a line of given length, whose extremities move 
in fixed lines in a given plane, describes an ellipse. 

Shew how to find the centre, and axes, of this ellipse j and 
the angular velocity about the centre of the ellipse of the tracing 
point when the describing line rotates uniformly. 

Transform this construction so as to shew that the ellipse is a 

6. A point. A, moves uniformly round one circular section of 
a cone; find the angular velocity of the point, a, in which the 
generating line passing through A meets a subcontrary section 
about the centre of that section. 

7. Solve, generally, the problem of finding the path by which a 
point will pass in the least time from one given point to another, 
the velocity at the point of space whose vector is p being expressed 
by the given scalar function y^. 

Take also the following particular cases : — 
(a.) fp=.a while Sap> 1, 
fp = h while Sap < 1 . 
{h.) fp = Sap. 
(c.) fp = -p^. (Tait, Trans. R. S. E., 1865.) 

8. If, in the preceding question,//) be such a function of Tp that 
any one swiftest path is a circle, every other such path is a circle, 
and all paths diverging from one point converge accurately in 
another. (Maxwell, Gam. and Bub. Math. Journal, IX. p. 9.) 


9. Interpret, as results of the composition of successive conical 
rotations, the apparent truisms 

y fi a 
and "^i -1^=1. 

Kid y p o. 

(Hamilton, Lectures, p. 334.) 

1 0. Interpret, in the same way, the quaternion operators 

} = (8s-')*(<f-')*(f»"')'. 

1 1 . rind the axis and angle of rotation by which one given rect- 
angular set of unit-vectors a, fi, y is changed into another given 
set Oi, Pi, yj. 

12. Shew that, if <f>p = p+ Vep, 

the linear and vector operation (^ denotes rotation about the vector e, 
together with uniform expansion in all directions perpendicular 
to it. 

Prove this also by forming the operator which produces the 
expansion without the rotation, and that producing the rotation 
without the expansion ; and finding their joint effect. 

13. Express by quaternions the motion of a side of one right 
cone rolling uniformly upon another which is fixed, the vertices of 
the two being coincident. 

14. Given the simultaneous angular velocities of a body about 
the principal axes through its centre of inertia, find the position 
of these axes in space at any assigned instant. 

15. Find the linear and vector function, and also the quaternion 
operator, by which we may pass, in any simple crystal of the 
cubical system, from the normal to one given face to that to an- 
other. How can we use them to distinguish a series of faces be- 
longing to the same zone ? 

16. Classify the simple forms of the cubical system by the 
properties of the linear and vector function, or of the quaternion 

17. Find the vector normal of a face which truncates symmetri- 
cally the edge formed by the intersection of two given faces. 

18. Find the normals of a pair of faces symmetrically truncating 
the g^ven edge. 


19. Find the normal of a lace which is equally inclined to three 
given faces. 

20. Shew that the rhombic dodecahedron may be derived from 
the cube, or from the octahedron, by truncation of the edges. 

2 1 . Find the form whose faces replace, symmetrically^ the edges 
of the rhombic dodecahedron. ♦ 

22. Shew how the two kinds of hemihedral forms are indicated 
by the quaternion expressions. 

23. Shew that the cube may be produced by truncating the edges 
of the regular tetrahedron. 

24. Point out the modifications in the auxiliary vector function 
required in passing to the pyramidal and prismatic systems re- 

25. In the rhombohedral system the auxiliary quaternion operator 
assumes a singularly simple form. Give this form, and point out 
the results indicated by it. 

26. Shew that if the hodograph be a circle, and the acceleration 
be directed to a fixed point ; the orbit must be a conic section, 
which is limited to being a circle if the acceleration follow any other 
law than that of gravity. 

27. In the hodograph corresponding to accelerationy(Z') directed 
towards a fixed centre, the curvature is inversely as D^y^D). 

28. If two circular hodographs, having a common chord, which 
passes through, or tends towards, a common centre of force, be cut 
by any two common orthogonals, the sum of the two times of hodo- 
graphically describing the two intercepted arcs (small or large) will 
be the same for the two hodographs. (Hamilton, Mements, p. 725.) 

29. Employ the last theorem to prove, after Lambert, that the 
time of describing any arc of an elliptic orbit may be expressed in 
terms of the chord of the arc and the extreme radii vectores. 

30. If $'( )s~^ be the operator which turns one set of rect- 
angular unit- vectors a, /3, y into another set oj, /3^, y^, shew that 
there are three equations of the form 



373.] We propose to conclude the work by giving a few in- 
stances of the ready appHcability of quaternions to questions of 
mathematical physics, upon which, even more than on the Geo- 
metrical or Kinematical applications, the real usefulness of the 
Calculus must mainly depend — except, of course, in the eyes of that 
section of mathematicians for whom Transversals and Anharmonic 
Pencils, &c. have a to us incomprehensible charm. Of course we 
cannot attempt to give examples in all branches of physics, nor 
even to carry very far our investigations in any one branch : this 
Chapter is not intended to teach Physics, but merely to shew by 
a few examples how expressly and naturally quaternions seem to be 
fitted for attacking the problems it presents. 

We commence with a few general theorems in Dynamics — the 
formation of the equations of equilibrium and motion of a rigid 
system, some properties of the central axis, and the motion of a solid 
about its centre of inertia. 

374.J When any forces act on a rigid body, the force /3 at the 
point whose vector is a, &c., then, if the body be slightly displaced, 
so that a becomes a + 6 a, the whole work done is 

This must vanish if the forces are such as to maintain -equilibrium. 
Henoe ike condition of equilibrium of a rigid body is 

2 SjSha = 0. 
For a displacement of translation 8a is ani/ constant vector, hence 

2/3 = (1) 

For a rotation-displacement, we have by § 350, e being the axis, 
and Ti being indefinitely small, 

6a = Ft a. 


and S/S.^Tea = S/S.fTa/S = S.eliFafi) = 0, 

whatever be e, hence 2 . Ta^ = (2) 

These equations, (1) and (2), are equivalent to the ordinary six 
equations of equihbrium. 

375.] In general, for any set of forces, let 
2/3 = /3i, 
2.ra/3 = ai, 

it is required to find the points for which the couple a-^ has its axis 
coincident with the resultant force ^^. Let y be the vector of such a 

Then for it the axis of the couple is 

2.F(a-y)^ = ai-ry^i, 
and by condition x^-^ = a^ — Fy/Sj . 

Operate by S^-^ ; therefore 

x^l ^ ^ai/3i, 
and Ty^i = a^ -ft-^^iA = -^Ja^^^-^, 

or y = ^«i/3r^+.5'i3i, 

a straight line (the Central Axis) parallel to the resultant force. 
376.] To find the points about which the couple is least. 
Here T{a^— Vyl3j) = minimum. 

Therefore S. (a^— FyjSj) F^iy = 0, 

where y' is any vector whatever. It is useless to try y'= ^^, but 
we may put it in succession equal to a^ and Vai^^. Thus 
S.yr.0^ra^P^ = Q, 
and {ra^^yf-fi\S.yra^p^ = 0. 

Hence y = x Va^ /Sj + j^/Sj , 

and by operating with S.Va^^^, we get 

or y= ra^Py-"^ +y/3i, 

the same locus as in last section. 
377.] The couple vanishes if 

«i- ^7/8i = 0. 
This necessitates Sa^fi^ = 0, 

or the force must be in the plane of the couple. If this be the case, 
still the central axis. 

224 QUATERNIONS. [378. 

378.] To assign the values of forces £, i^, to act at «, ej, and be 
equivalent to the given system. 

Hence Fe^H- n^ifi^-i) = a^, 

and i = (e- ei)-i (a^ - Tei ^1) + a; (e - €1). 

Similarly for f^. The indefinite terms may be omitted, as they 
must evidently be equal and opposite. In fact they are any equal 
and opposite forces whatever acting in the line joining the given 
. 379.] For the motion of a rigid system, we have of course 

^S{md—/3)ba = 0, 
by the general equation of Lagrange. 

Suppose the displacements 6a to correspond to a mere translation, 
then 8a is an^ constant vector, hence 

'2{md — 0) = 0, 
or, if ai be the vector of the centre of inertia, and therefore 

a^'Em = 'Ema, 
we have at once di'Sm — 2/3 = 0, 

and the centre of inertia moves as if the whole masa were concen- 
trated in it, and acted upon by all the applied forces. 

380.] Again, let the displacements 8 a correspond to a rotation 
about an axis «, passing through the origin, then 

ba = Fea, 
it being assumed that Te is indefinitely small. 

Hence I,S.eFa{m'd—^) = 0, 

for all values of e, and therefore 

I,.Fa{md-0) = 0, 
which contains the three remaining ordinary equations of motion. 

Transfer the origin to the centre of inertia, i. e. put a = a^ + ot, 
then our equation becomes 

2r(a, + in-) (jKiii + »««■— /3) = 0. 
Or, since 2»»ot = 0, 

2 Fot (»» OT - y3) + Fai(ai 2 J»- 2/3) = 0. 
But aj2»»— 2/3 = 0, hence our equation is simply 

^V'mimih-^) = 0. 
Now 2Fi!r/3 is the couple, about the centre of inertia, produced 
by the applied forces ; call it (/>, then 

ImFs^ii = <{) (1) 


381 .] Integrating once, • 

I.mF'ST^ = y+/<f)di (2) 

Again, as the motion considered is relative to the centre of inertia, 
it must be of the nature of rotation about some axis, in general 
variable. Let e denote at once the direction of, and the angular 
velocity about, this axis. Then, evidently, 

•a = Vetss. 

Hence, the last equation may be written 
'S.mzrYiTS = yJrf^dt. 
Operating by S.i, we get 

2m{Fem)^ = Sey + Se/<f,dt (3) 

But, by operating directly by 2fSidt upon the equation (1), we get 

2»?(reCT)2 =-h^ + 2fSi<i>dt (4) 

(2) and (4) contain the usual four integrals of the first order. 

382.] When no forces act on the body, we have ^ = 0, and 

therefore '2,mw Few = y, (5) 

Imir^ = ■2miFi'!:Tf = —A^, (6) 

and, from (5) and (6), Sey =—Jfi (7) 

One interpretation of (6) is, that the kinetic energy of rotation 
remains unchanged : another is, that the vector e terminates in an 
ellipsoid whose centre is the origin, and which therefore assigns 
the angular velocity when the direction of the axis is given ; (7) 
shews that the extremity of the instantaneous axis is always in 
a plane fixed^in space. 

Also, by (5), (7) is the equation of the tangent plane to (6) at 
the extremity of the vector e. Hence the ellipsoid (6) rolls on the 
plane (7). 

From (5) and (6), we have at once, as an equation which e must 
satisfy, y2 2.^ ( FimY= —k^ (2.»8sr Fivrf. 

This belongs to a cone of the second degree fixed in the body. Thus 
all the ordinary results regarding the motion of a rigid body under 
the action of no forces, the centre of inertia being fixed, are deduced 
almost intuitively : and the only difficulties to be met with in more 
complex properties of such motion are those of integration, which 
are inherent to the subject, and appear whatever analytical method 
is employed. (Hamilton, Proc. B. I. A. 1848.) 

383.] Let a be the initial position of ■nr, q the quaternion by 
which the body can be at one step transferred from its initial posi- 
tion to its position at time t. Then 

ra- = qaq~^ 


and Hamilton's equation (5) of last section becomes 

or ^.mq {aS.aq~^ tq—q~^(qa? } q'"^ = y. 

Let <^p = 'Si.m{a8ap—a?p), (1) 

where is a self-conjugate linear and vector function, whose con- 
stituent vectors are fixed in the body in its initial position. Then 
the previous equation may be written 

or </>(S'~^«S') = rVS'- 

For simplicity let us write 

r^'i = r),^ 

Then Hamilton's dynamical equation becomes simply 

0'? = C. (3) 

384.3 ^^ is ®^y *o s^^ what the new vectors r\ and ( represent. 
For we may write (2) in the form 

e = qm-\ \ (2') 

from which it is obvious that rj is that vector in the initial position 
of the body which, at time t, becomes the instantaneous axis in the 
moving body. When no forces act, y is constant, and f is the 
initial position of the vector which, at time t, is perpendicular to 
the invariable plane. 

385.] The complete solution of the problem is contained in equa- 
tions (2), (3) above, and (4) of § 356*. Writing them again, we 

qr)=M, (4) 

7i = iC, (2) 

0'? = f. (3) 

We have only to eliminate f and »;, and we get 

2q = q<f>-^q-^yq), (5) 

in which q is now the only unknown ; y, if variable, being supposed 
known in terms of q and t. It is hardly conceivable that any 
simpler, or more easily interpretable, equation for q can be presented 

* To these it is unnecessary to add 

Z'g= constant, 
as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there 
must have been a quaternion in the place of the vector i/. In &ct, 

^(Tqr = 2S.qKq^{Tqf8n'=0. 


until symbols are devised far more comprehensive in their meaning 
than any we yet have. 

386.] Before enfering into considerations as to the integration 
of this equation, we may investigate some other consequences of 
the group of equations in § 385. Thus, for instance, differentiating 
(2), we have 

and, eliminating q by means of (4), 

yqri + 2yq = qt,C+2qC, 

whence C=yCn+ i~^ yq. ; 

which gives, in the case when no forces act, the forms 

t=yp^-H, (6) 

and (as C= ^^) 

<l>ri= — F.ri<t>ri (7) 

To each of these the term q~^ yq, or q~^ yjfq, must be added on the 
right, if forces act. 

387.] It is now desirable to examine the formation of the fanc- 
tion <f). By its definition (1) we have 

<l>p = 2.M (aSap — a^p), 
= — 'S.maVap. 
Hence —Sp(Pp = 'S,.m{Trapf, 

so that — Sp<pp is the moment of inertia of the body about the 
vector p, multiplied by the square of the tensor of p. Thus the 
equation g^^p ^ _p^ 

evidently belongs to an ellipsoid, of which the radii-vectores are 
inversely as the square roots of the moments of inertia about them ; 
so thatj if i, j, k be taken as unit- vectors in the directions of its 
axes respectively, we have 

Si<j)i = — A, \ 

Sj<f>j=-BA (8) 

Sk<t)k = -C,) 
A, B, C, being the principal moments of inertia, Consequently 

4>p = —{AiSip + £JSjp+ CkSip} (9) 

Thus the equation (7) for rj breaks up, if w^ put 

into the three following scalar equations 

Aa)i+ (C— 5)q)2C»3 = 0, j 
Sd}^ + {A — C) w^coj^ = 0, I 
C(02 + {B — A) o>^a>2 = 0, ) 
Q 2 

228 QUATEENIONS. [388. 

which are the same as those of Euler. Only, it is to be understood 
that the equations just written are not primarily to be considered 
as equations of rotation. They rather expres* with reference to 
fixed axes in the initial position of the body, the motion of the 
extremity, toj, Ug, (1)3, of the vector corresponding to the instan- 
taneous axis in the moving body. If, however, we consider tOj, Wg, cog 
as standing for their values in terms of w, x,y, «: (§ 391 below), or 
any other coordinates employed to refer the body to fixed axes, they 
are the equations of motion. 

Similar remarks apply to the equation which determines f, for if 
we put f=i^^ + y^^ + ^^^^ 

(6) may be reduced to three scalar equations of the form 

''^9'^^ = 0. 

388.] Euler's equations in their usual form are easily deduced 
from what precedes. For, let 

whatever be p ; that is, let + represent with reference to the moving 
principal axes what ^ represents with reference to the principal 
axes in the initial position of the body, and we have 
<t.e = q^ (q-^ iq) q'^ = q<l> (n) q'^ 

= qiq-' =qr{C'l>-H)q-' 

= -qr{ri<t>ri)q-^ 

. =-V.qri<p{n)q-'^ 

= -r.qr,q-'^q(t){q-'^eq)q-^ 

which is the required expression. 

But perhaps the simplest mode of obtaining this equation is to 
start with Hamilton's unintegrated equation, which for the case 

of no forces is simply 

S.»«FisrOT = 0. 

But from ot =: Vezr 

we deduce «• ?= Fe^+ Vk-sr 

= ore^ — e<S«CT+ Vkvr, 

so that 2.«M(F'e«riS€OT — eCT^ + cr^eBr) = 0. 

If we look at equation (1), and remember that ^ differs from 

simply in having ot substituted for a, we see that this may be 

written Fe+e + ^e = 0, 


the equation before obtained. The first mode of arriving at it has 
been given because it leads to an interesting set of transformations, 
for which reason we append other two. 

By (2) y = qCq-\ 

therefore = qq-'^.q^q-'^+q^q-'^—q^^q-'^q^'^, 

or q^q-^ = iV.yVqq-'^ 

= Fye. 
But, by the beginning of this section, and by (5) of § 382, this 
is again the equation lately proved. 

Perhaps, however, the following is neater. It occurs in Hamil- 
ton's Elements. 

By (5) of §382 +€ = y. 
Hence <t>e =—<}>«=— S.w(t3- Few + ot Fenr) 

= — 'Si.m'iiSesi 
= — F'.f'2.m'srSe'ST 
= - re4.e. 
389.] However they are obtained, such equations as those of 
§ 387 were shewn long ago by Euler to be integrable as follows. 

letting 2fm^<i,^mjt = s, 

we have j^^z =JQ^^ + (£- C) s, 

with other two equations of the same form. Hence 

2dt=: - 

so that t is known in terms of s by an elliptic integral. Thus, 
finally, tj or f may be expressed in terms of i ; and in some of the 
succeeding investigations for q we shall suppose this to have been 
done. It is with this integration, or an equivalent onCj that most 
writers on the farther development of the subject have commenced 
their investigations. 

390.] By § 381, y is evidently the vector moment of momentum 
of the rigid body ; and the kinetic energy is 

But Sey = S.q-^eqq~^yq = SrjC 

so that when no forces act 

But, by (2), we have also 

TC=Ty, or T<f>r, = Ty, 
so that we have, for the equations of the cones described in the 

230 QUATERNIONS. [39 1. 

initial position of the body by rj and t, that is, for the cones de- 
scribed in the moving body by the instantaneous axis and by the 
perpendicular to the invariable plane, 

This is on the supposition that y and & are constants. If forces act, 
these quantities are functions of t, and the equations of the cones 
then described in the body must be found by eliminating t between 
the respective equations. The final results to which such a process 
will lead must, of course, depend entirely upon the way in which t 
is involved in these equations, and therefore no general statement 
on the subject can be made. 

391.] Recurring to our equations for the determination of q, and 
taking first the case of no forces, we see that, if we assume tj to 
have been found (as in § 389) by means of elliptic integrals, we have 
to solve the equation „ .^ 

that is, we have to integrate a system of four other difiPerential 
equations harder than the first. 

Putting, as in § 3 8 7, n = icOj^ +j\ + kw^ , 

where Wj, Wg, W3 are supposed to be known functions of t, and 

q = w+ico + jy + kz, 

... , . \ ,, dm dx du dz 

this system IS -di = ^ = y = Y ~ 'Z' 

* To get an idea of the nature of this equation, let us integrate it on the supposi- 
tion that ij is a constcmt vector. By differentiation and substitution, we get 

Hence „_ «,.«=. ^^ * j_ n =i- ^^ t 

g= ^icos — « + QsSin^ t. 

Substituting in the given equation we have 

2^ C^ e, sin 2l e +& cos ^ «) = («, cos ^ «+ e, sin ^ ^j-J- 

Hence Tiy.Ga = Q, 1;, 

which are virtually the same equation, and thus. 

And the interpretation of 2 ( ) q~^ will obviously then be a rotation about ij through 
the angle tTrj, together with any other arbitrary rotation whatever. Thus any posi- 
tion whatever may be taken as the initial one of the body, and Q, ( ) Q,-» brings it 
to its required position at time < = 0. 


where ^= — <o,a;— Wgy — ojj^, 

X= Wj^W + tBg^ — a^i^, 

^= w^w + a^se — ooi^l 
or, as suggested by Cayley to bring out the skew symmetry, 
X= . (ja^y — oi^z + ai-^w, 
T=.—m^x . + a-j^z + (o^w, 
Z ■=■ oj^a; — Wj^y . -^m^w, 

W ■=—<it>-^X — ai2,y — <«>3« . 
Here, of course^ one integral is 

w^ +(xi'^ +^^+z^ = constant. 
It may suffice thus to have alluded to a possible mode of solution, 
which, ^except for very simple values of ri, involves very great diffi- 
culties. The quaternion solution, when rj is of constant length and 
revolves uniformly in a right cone, will be given later. 

392.] If, on the other hand, we eliminate t], we have to inte- 
grate S^~^ir^72)=^i' 

so that one integration theoretically suffices. But, in consequence 
of the present imperfect development of the quaternion calculus, the 
only known method of effecting this is to reduce the quaternion 
equation to a set of four ordinary differential equations of the first 
order. It may be interesting to form these equations. 
Put q = w+iai + jy + iz, 

Y = ia+Jb + ^o, 
then, by ordinary quaternion multiplication, we easily reduce the 
given equation to the following set : 

di d/w dx dy dz 


W= — x'^—y3&—ze. or X= . yC— «13+wa, 

x= wa+^ffi— ^B r=— ««[; . +z^+wii, 

T= w'Q+z%—ui!<S; z= !JBi&—y% . +w(i::, 


a = -J [a (w^— a;'' —y^ —z^) + 2a? {m + hy -\-ez) + 2w {bz—cy)'], 
33 = -^ [5 {w^ —afi —y"^ —z^) + ly {ax + by + cz) + 2w {cx—az)'], 
a: = -^ [c {w^ —x^ —y^—z^) + 2z{ax+6y + cz) + 2w {ay-^bx)], 

232 QUATERNIONS. [393« 

JF, X, Y, Zare thus homogeneom functions of w, x, y, z of the third 

Perhaps the simplest way of obtaining these equations is to trans- 
late the group of § 385 into w, x, y, z at once, instead of using the 
equation from which f and r\ are eliminated. 

We thus see that ^ ^ *a+yi8 +/^ffi. 

One obvious integral of these equations ought to be 
vfi + x"^ +y^ +z^ = constant, 
which has been assumed all along. In fact, we see at once that 

identically, which leads to the above integral. 

These equations appear to be worthy of attention, partly because 
of the homogeneity of the denominators W, X, T, Z, but particularly 
as they afford (what does not appear to have been sought) the means 
of solving this celebrated problem at one step, that is, without the 
previous integration of Euler's equations (§ 387). 

A set of equations identical with these^ but not in a homogeneous 
form (being expressed, in fact, in terms of k, \, |u, v of § 359, instead 
of 10, x,y, z), is given by Cayley {Gamb. and Bub. Math. Journal, 
vol. i. 1846), and completely integrated (in the sense of being re- 
duced to quadratures) by assuming Euler's equations to have been 
previously integrated. (Compare § 391.) 

Cayley's method may be even more easily applied to the above 
equations than to his own ; and I therefore leave this part of the 
development to the reader, who will at once see (as in § 391) that 
%, 38, ffi correspond to coi, Wg, tag of the rj type, § 387. 

393.J It may be well to notice, in connection with the formulae 
for direction cosines in § 359 above, that we may write 

% = --j\a{:w'^-\-x^—y'''—z^)-^il{xy + 'wz)-\-'ic{pz—wy)'], 
38 = -^\2a{xy — wz)-{h(vP-—x'^->ry'^—z''-)-\-1c{yz-\-wxy\, 

(!t = -p^[2a(xz + wy) + 2b {yz—wx) + c {w^ —x^ —y"^ + z^)']. 

These expressions may be considerably simplified by the usual 
assumption, that one of the fixed unit- vectors {i suppose) is perpen- 
dicular to the invariable plane, which amounts to assigning defi- 
nitely the initial position of one line in the body ; and which gives 
the relations 5—0 c = 



394.] Wlieii forces act, y is variable, and the quantities a, h, c 
will in general involve all the variables w, x, y, z, t, so that the 
equations of last section become much more complicated. The type, 
however, remains the same if y involves t only ; if it involve q we 
must differentiate the equation, put in the form 

and we thus easily obtain the differential equation of the second 
order ^ = iV.qct) (q-^ q) q-^ + 2 qcj) {F. q-^q) q-^ ; 

if we recollect that, because q~^q is a vector, we have 

Though remarkably simple, this formula, in the present state of 
the development of quaternions, must be looked on as intractable, 
except in certain very particular cases. 

395.] Another mode of attacking the problem, at first sight 
entirely different from that in § 383, but in reality identical with 
it, is to seek the linear and vector function which expresses the 
Homogeneous Strain which the body must undergo to pass from its 
initial position to its position at time t. 

Let -ST = xfflj 

a being (as in § 383) the initial position of a vector of the body, 
■ST its position at time t. In this case x i^ ^ linear and vector 
function. (See § 360.) 

Then, obviously, we have, ^-^ being the vector of some other point,, 
which had initially the value a^, 

Siss'ST^ = S.\a)(a.i = Saa^, 
(a particular case of which is 

T'ST = ^xa = Ta) 
and Fototj = J^-x^X^^i = x^"«i' 

These are necessary properties of the strain-function x, depending 
on the fact that in the present application the system is rigid. 

396.] The kinematical equation 

CT = Few 
becomes Xa = F. exa 

(the function x being formed from x by the differentiation of its 
constituents with respect to t). 

Hamilton's kinetic equation 

S.warFera- = y, 
becomes 'Si.mxaF.exa = y. 

234 QUATERNIONS. [39 7> 

This may be written 

2.««(xaiS'.€xa— eo^) = V) 

or I,.m{aS.ax'f-x~^e.a^) = X~^Y> 

where x' is the conjugate of x- 

But, because '^•X'^X'h. = '^""u 

we have Saa^ = S.ax'xa^, 

whatever be a and a^, so that 

X = X ^• 
Hence 2.m{w8.ax~^e—x~^e.a^) = x~V, 

or, by §383, ^^-i^^^-iy^ 

397.] Thus we have, as the analogues of the equations in 
§§ 383, 384, ^-1^ ^ ^^ 

x-V = C, 
and the former result x" = ^' «X° 

becomes X** =^'X'7X'* = X^**- 

This is our equation to determine X) V being supposed known. 
To find rj we may remark that 

<f>l = C, 

and C = X~V- 

But XX~^« = a. 

so that XX~^« + XX~'« = 0. 

Hence f=-X~^XX~V 

= -r.r,x-^Y=J'Cv=^-C4>-'C 
or </>^ = — Ftj^tj. 

These are the equations we obtained before. Having found rj 
from the last we have to find x from the condition 


398.] We might, however, have eliminated ?j so as to obtain an 
equation containing x a^lone, and corresponding to that of § 385. 
For this purpose we have 

jj = ,^-if= ^-^x"^y> 

so that, finally, X~^X'"' = ^- 't>~^ X~''>"»> 

or X~^ « = ^' X" ^ <'0~^X~^y' 

which may easily be formed from the preceding equation by putting 

X~^a for a, and attending to the value of x"^ given in last section. 



399.] We have given this process, though really a disguised form 
of that in §§ 383, 385, and though the final equations to which 
it leads are not quite so easily attacked in the way of integration as 
those there arrived at, mainly to shew how free a use we can make 
of symholic functional operators in quaternions without risk of 
error. It would be very interesting, however, to have the problem 
worked out afresh from this point of view by the help of the old 
analytical methods : as several new forms of long-known equations, 
and some useful transformations, would certainly be obtained. 

400.] As a verification, let us now try to pass from the final 
equation, in x alone, of § 398 to that of § 385 in ^ alone. 

We have, obviously, 

OT = qa£r^ = X«. 
which gives the relation between q and x- 

[It shews, for instance, that, as 

yS.^Xa = -S.ax'A 
while 'S-zSxa = S.^qaq-'^ = S.aq-^^q, 

we have x'/3 = T'^Pi^ 

and therefore that xx'i^ = id'^Pi)^^ = i^, 

or x' = X~^j ^s above.] 

Difierentiating, we have 

qaq^'^—qaq'^qq'^ = x«- 
Hence X'^X" = S'~^?<*~"2^*? 

= 2r.r{q-'^q)a. 

Also ^~^X~V = ^"^(^^V?). 

so that the equation of § 398 becomes 

2r.r{q-'^q)a= V. (^-^ {q-^7q) a, 

or, as a may have any value whatever, 

2r.q'^q = ^-Hq-^yq), 

which, if we put Tq = constant 

as was originally assumed, may be written 

2q = q<l>-\q-^yq), 
as in § 385. 

401.] To form the equation for Precession and Nutation,. Let o- 
be the vector, from the centre of inertia of the earth, to a particle 
m of its mass : and let p be the vector of the disturbing body, whose 
mass is M. The vector-couple produced is evidently 


= M^. 


no farther terms being necessary, since =- is always small in the 

actual cases presented in nature. But, because o- is measured from 
the centre of inertia, S.?»o-= 0. 

Also, as in § 383, <^p = 2.«! {aScrp—tr'^p). 

Thus the vector-couple required is 

Referred to coordinates moving with the body, ^ becomes 4> as in 
§ 388, and § 388 gives 



Simplifying the value of <|> by assuming that the earth has two 
principal axes of equal moment of inertia, we have 

Bf—{A—B)aSaf = vector-constant + ZM{A—B) / ^g °^ 


This gives Sat = const. = i2, 

whence e = — i2a -|- act, 

so that, finally, 

BVad-Aaa = ^{A-B)rap8ap. 

The most striking peculiarity of this equation is that Reform of 
the solution is entirely changed, not modified as in ordinary cases 
of disturbed motion, according to the nature of the value of p. 

Thus, when the right-hand side vanishes, we have an equation 
which, in the case of the earth, would represent the rolling of a 
cone fixed in the earth on one fixed in space, the angles of both 
being exceedingly small. 

If p be finite, but constant, we have a case nearly the same as 
that of a top, the axis on the whole revolving conically about p. 


But if we assume the expr *sion 

p = r{Jeosmt + k sin mt)j 
(which represents a circular orbit described with uniform velocity,) 
a revolves on the whole conically about the vector i, perpendicular 
to the plane in which p lies. {Trans. B, 8. E., 1868-9.) 

402.] To form the eq%iation of motion of a simple 'pendulum, 
taking account of the eartVs rotation. Let a be the vector (from 
the earth's centre) of the point of suspension, X its inclination to 
the plane of the equator, a the earth's radius drawn to that point ; 
and let the unit-vectors i, j, h be fixed in space, so that i is parallel 
to the earth's axis of rotation ; then, if m be the angular velocity 
of that rotation 

a = « p sin A + (/ cos 01^ + ^ sin ad) cos A] (1) 

This gives a = a o) ( —j sin tu^ + A cos mf) cos \ 

^ = inYia ...(2) 

Similarly a = m Yia = — o)^ (a — ai sin A) (3) 

403.] Let p be the vector of the bob m referred to the point of 
suspension, R the tension of the string, then if oj be the direction 

ofpuregravity m{d + p) =-mgUay-BUp, (4) 

which may be written 

rpd+rpp = ^ja,p (5) 

To this must be added, since r (the length of the string) is constant, 

Tp = r, (6), 

and the equations of motion are complete. 

404.] These two equations (5) and (6) contain every possible case 
of the motion, from the most infinitesimal oscillations to the most 
rapid rotation about the point of suspension, so that it is necessary 
to adapt different processes for their solution in different cases. 
We take here only the ordinary Foucault case, to the degree of 
approximation usually given. 

405.] Here we neglect terms involving m^. Thus we write 

a = 0, 
and we write a for Oj , as the difference depends upon the ellipticity 
of the earth. Also, attending to this, we have 


p= — -a + i!T, (7) 

whereby (by (6)) xSoot = 0, (8) 

and terms of the order ot^ are neglected. 

238 QUATERNIONS. [405. 

With (7), (5) becomes 

— — Vwss = — Foot ; 
a a 

so thatj if we write -■=•«?, (9) 

we have FaC* + w^ot) = (10) 

Now, the two vectors ai— asia\ and Via 

have, as is easily seen, equal tensors ; the first is parallel to the line 
drawn horizontally northwards from the point of suspension, the 
second horizontally eastwards. 

Let, therefore, w = «;(«»- o sin A) +j^ria, (11) 

which {x and y being very small) is consistent with (6). 

From this we have (employing (2) and (3), and omitting a?) 
•is = cb {ai— asinX) + yFia—xm ainXFia— yo) {a— ai sin \), 
a =z x{ai — aaiaK)+ifFia—2dia>BmKFia—2ya){a—aiBia\). 
With this (10) becomes 
Fa[ai(aJ— a sin \) + yFia—2xoi s\n\Fia—2ym{a—ai sin \) 

+ n^x{ai—asm\)^n^yFia] = 0, 
or, if we note that F. a Fia = a{ai—a sin \), 

(^—x—2ya>smk—n,^x)aFia + {t/ — 2ii;a)8in.k + n'^y)a(ai—asm\) = 0. 
This gives at once x + n^x+ 2a>jfsm\ = 0, 

which are the equations usually obtained ; and of which the solution 
is as follows : — 

If we transform to a set of axes revolving in the horizontal plane 
at the point of suspension, the direction of motion being from the 
positive (northward) axis of x to the positive (eastward) axis of y, 
with angular velocity ii, so that 

a; = f cos Slt—r) sin Sit, 
^ = f sin Qft + t) cos 12 1, 
and omit the terms in D? and in w 12 (a process justified by the 
results, see equation (15)), we have 
({+«^0 cos Q,t-(ij + n^ri) sin Q.t-2^ {il—co sinX) = 0, ) 

So that, if we put il = oism\, (15) 

we have simply f +*^£ = 0, ) 

ij + n''r, = 0j ^^"^ 

the usual equations of elliptic motion about a centre of force in the 
centre of the ellipse. (Proc. E. S. K, 1869.) 

=::} <-> 

';} <"' 


406.] To construct a reflecHkig surface from which rays, emitted 
from a point, shall after reflection diverge uniformly, hut horizontally. 

Using the ordinary property of a reflecting surface, we easily 
obtain the equation 

S.dp{^±^% = Q. 

By Hamilton's grand Theory of Systems of Bays, we at once write 
down the second form 

Tp—T(fi+aFap) = constant. 

The connection between these is easily shewn thus. Let ot and 
T be any two vectors whose tensors are equal, then 

whence, to a scalar factor ^re*, we have 

\i T + 'S!- 



Hence, putting w = C/'(/3 + aVap) and r = Up, we have from the first 
equation above 

S.dplUp+ Ui^ + aVap)'] = 0. 

But d(p + aFap) = aVadp =—dp—aSadp, 

and S.a(fi + a Vap) = .0, 

so that we have finally 

S.dpUp-S.d{^ + aFap)U{^ + arap) = 0, 
which is the differential of the second equation above. A curious 
particular case is a parabolic cylinder, as may be easily seen geo- 
metrically. The general surface has a parabolic section in the plane 
of a, y3 ; and a hyperbolic section in the plane of /3, a0. 

It is easy to see that this is but a single case of a large class of 
integrable scalar functions, whose general type is 

S.dp(^'p = 0, 

the equation of the reflecting surface ; while 

8{<T—p)dcT — 
is the equation of the surface of the reflected wave : the integral of 
the former being, by the help of the latter, at once obtained in the 
form Tp + ^(a—p) = constant*. 

407.] We next take Fresnel's Theory of DouMe Refraction, but 

* Proe. R. S. E., 1870-71. 

240 QUATERNIONS. [408. 

merely for the purpose of shewing how quaternions simpHfy the 
processes required, and in no way to discuss the plausibility of the 
physical assumptions. 

Let tzT he the vector displacement of a portion of the ether, with 
the condition ^2 __i /j\ 

the force of restitution, on Fresnel's assumption, is 
tiflHSvar + b^jSj':!T + c^kSkin) = t<fm, 
using the notation of Chapter V. Here the function <^ is obviously 
self-conjugate, a^, b^, c^ are optical constants depending on the 
crystalline medium, and on the colour of the lightj and may be 
considered as given. 

Fresnel's second assumption is that the ether is incompressible, 
or that vibrations normal to a wave front are inadmissible. If, then, 
a be the unit normal to a plane wave in the crystal, we have of 

course a^=-\, (2) 

and Six's! = 0; (3) 

but, and in addition, we have 

■s!~^ Vtz^Ts II a, 
or S.aTu^ = (4) 

This equation (4) is the embodiment of Fresnel's second assumption, 
but it may evidently be read as meaning, the normal to the front, the 
direction, of vibration, and that of the force of restitution are in one 

408.] Equations (3) and (4), if satisfied by -m, are also satisfied 
by Tsa, so that the plane (3) intersects the cone (4). in two lines 
at right angles to each, other. That is, for any given wave front 
there are two directions of vibration, and they are perpendicular to each 

409.] The square of the normal velocity of propagation of a plane 
wave is proportional to the ratio of the resolved part of the force of 
restitution in the direction of vibration, to the amount of displace- 
ment, hence j;2 = S-as^Tn, 
Hence Fresnel's Wave-surface is the envelop of the plane 

Sap ^ i\/Sm<^, (5) 

with the conditions vt^ = — \, (1) 

a''=-l, (2) 

Sour =0, (3) 

S. aiJ7<l)'ar = (4) 


Formidable as this problem appears, it is easy enough. From (3) 
and (4) we get at onee^ 

Henee^ operating by S. ct, 

— CO ^ — S'STcfyar = — v^. 
Therefore ((jb + »2) ^ = _ a^ac^^-, 

and S.a {(j) + v^)-^ a = (6) 

In passing, we may remark that tMs equation gives the normal velo- 
cities of the two rays whose fronts are perpendicular to a. In Cartesian 
coordinates it is the well-known equation 

P wfi rfi _ 

a^—v^ ■*" P3^2 + ^2~^ = °- 
By this elimination of or, our equations are reduced to 

S.a{(i> + v^)-'^a= 0, (6) 

V zzz-Sap, (5) 

a^ =-1 (2) 

They give at once, by § 309, 

{ct> + v^)-^a + vpSa{cj>-{-v^)-^a = ha. 
Operating by S.a we have 

v^Sa{<tj + v^)-^a = h. 
Substituting for h, and remarking that 

Sa{(t> + v^)-^a =-T^{(j> + t)2)-i a, 
because <^ is self-conjugate, we have 

/J . 2\-i va — p 

p^ + v^ 
This gives at once, by rearrangement, 

^{(l> + v^)-^a = {<t>-p^)-Y 
Hence {<t>-p^)-^P = ^^^ ■ 

Operating by S.p on this equation we have 

Sp{<P-p')-^p = -l, (7) 

which is the required eqjflation. 

[It will be a good exercise for.the student to translate the last 
ten formulae into Cartesian coordinates. He will thus reproduce 
almost exactly the steps by which Archibald Smith * first arrived 
at a simple and symmetrical mode of .effecting the elimination. Yet, 
as we shall presently see, the above process is far from being the 
shortest and easiest to which quaternions conduct us.J 

* Cambridge Phil. Trans., 1835. 


410.] The Cartesian form of the equation (7) is not the usual 
one. It is, of course, 

aj2 yi g^ 

But write (7) .in the form 

and we have the usual expression 

a2^2 ^2 ,,2 „2,2 

I 7,2 „2 T .!! ..•>. " 

ast quaternion eqi 

This last quaternion equation can also be put into either of the new 

or 2'(p-2-,^-i)-4p = 0. 

411.] By applying the results of §§ 171, 172 we may introduce 
a multitude of new forms. We must confine ourselves to the most 
simple ; but the student may easily investigate others by a process 
precisely similar to that which follows. 

Writing the equation of the wave as 

where we have g = — p~^, 

we see that it may be changed to 

if mSp<f>p = ffkp^ ■=—h. 

Thus the new form is ^ 

Sp{(j)-^—mSp(l)p)-^p = (1) 

Here m = -^^^ , 8p^p = a^ap' + V^y"^ + c^z^, 

and the equation of the wave in Cartesian coordinates is, putting 

' ^ + -...« ... = 0. 

412.] By means of equation (1) of last section we may easily 
prove Pliicker's Theorem, The Wave-Surface is its own reciprocal with 
resjieci to the ellipsoid, whose equation is 

Sp^^p = —7— • 


The equation of the plane of contact of tangents to this surface from 
the point whose vector is p is 

iSWd)* p = —, — 

The reciprocal of this platie, with respect to the unit-sphere about 
the origin, has therefore a vector cr where 

a = \/m,(ji^ p. 
Hence p = —t— (b~^a; 

and when this is substituted in the equation of the wave we have 
for the reciprocal (with respect to the unit-sphere) of the reciprocal 
of the wave with respect to the above ellipsoid, 

S.cr (^ - — Sacj)-^ 0-) 0- = 0. 

This differs from the equation (1) of last section solely in having 

(p~^ instead of (f>, and (consistently with this) — instead of m. Hence 

it represents the index-surface. The required reciprocal of the wave 
with reference to the ellipsoid is therefore the wave itself. 

413.J Hamilton has given a remarkably simple investigation of 
the form of the equation of the wave-surface, in his Elements, p. 736, 
which the reader may consult with advantage. The following is 
essentially the same, but several steps of the process, which a skilled 
analyst would not require to write down, are retained for the benefit 
of the learner. 

Let %= — 1 (1) 

be the equation of any tangent plane to the wave^ i.e. of any wave- 
front. Then /u is the vector of wave-slowuess, and the normal 

velocity of propagation is therefore -=p- . Hence, if isr be the vector 

direction of displacement, ju~^«r is the effective component of the 
force of restitution. Hence, ^w denoting the whole force of re- 
stitution, we have ^'sr—pr^'oi || p., 

or -m II {4>—ijr^)-^p., 

and, as ss is in the plane of the wave-front, 

Sp.'d = 0, 

or SiJi.{(f)-p.-'')-^iJ. = (2) 

This is, in reality, equation (6) of § 409. It appears here, how- 
ever, as the equation of the Index-Surface, the polar reciprocal of 

E % 

244 QUATERNIONS. [4 1 4. 

the wave with respect to a unit-sphere about the origin. Of course 
the optical part of the problem is now solved, all that remains being 
the geometrical process of § 3 1 1 . 

414.] Equation (2) of last section may be at once transformed, 
by the process of § 410, into 


Let us employ an auxiliary vector 

whence ij,= (jx'^—(J)-^)t (1) 

The equation now becomes 

Sh.t=1, (2) 

or, by (1), y?T-^-S!r4r'^T = 1 .- (3) 

Differentiating (3), subtract its half from the result obtained by 
operating with S.t on the differential of (1). The remainder is 

T'^Sixdn—STdjj. = 0. 
But we have also (§311) Spdix = 0, 
and therefore xp = jxt^—t, 

where a; is a scalar. 

This equation, with (2), shews that 

Stp = (4) 

Hence, operating on it by S.p, we have by (1) of last section 

xp^ = — r^, 
and therefore p~^ =— /x + r"^. 

This gives p~^ = ij,^ — t~'^. 

Substituting from these equations in (1) above, it becomes 

or r = ((^~^— p~^)~^p~^. 
Finally, we have for the required equation, by (4), 
^p-i(<^-i_p-2)-ip-i = 0, 
or, by a transformation already employed, 
415,] It may assist the student in the practice of quaternion 
analysis, which is our main object, if we give a few of these invest- 
igations by a somewhat varied process. 
Thus, in § 407, let us write as in § 168, 

aHSv^Jfl^jSj-ss^c^hSk^ = yxS/OT + Z/SW-yOT. 
We have, by the same processes as in § 407, 

S.VTaX'Si/t!r + S.'maix'Sk''!!T = 0. 





This may be written, so flr as the generating lines we require are 

since -sra is a vector. 
Or we may write 

S.[l,'V.'7T\'-S!a = = /S./yl'OTX'OTa. 

Equations (1) denote two cones of the second order which pass 
through the intersections of (3) and (4) of § 407. Hence their in- 
tersections are the directions of vibration. 

416.] By (1) we have 

S.T!TX.''sraix'= 0. 
Hence ■nrX'tn-j a, \i.' are coplanar ; and, as tn- is perpendicular to a, it 
is equally inclined to Vk'a and Fix a. 

For, i£ L, M, A be the projections of k', f/, a on the unit 
sphere, £C the g-reat circle whose 
pole is A, we are to find for the 
projections of the values of w^ on 
the sphere points P and P', such 
that if LF be produced till 

Q may lie on the great circle AM. 
Hence, evidently, 

CP = PB, 
and C^F=rB; 
which proves the proposition, since 
the projections of Vk'a and Vj/a on the sphere are points b and 
c in BC, distant by quadrants from C and B respectively. 
417.] Or thus, Svra = Q, 

S.srV.ak'-snx — 0, 

therefore as'sr = F. a K ak''as-ii, 

= - r. W/ -aSaF. W/x'. 

Hence {Sk'ix-a;) ot = (X' + aSak') ^/x'w + {/ + aSaf/) Sk'w. 

Operate by S.k', and we have 

(x + Sk'aSi/a) Sk'^ = [X'^ a^-S'^ X'a] -S/xV 

= Si/^T'^Fk'a. 
Hence by symmetry, 




246 QUATERNIONS. [4 1 8. 

"'' T7k'a - TFi/a - ' 

and as fco = 0, 

418.] The optical interpretation of the common result of the 
last two sections is that the planes of polarization of the two rays 
whose wave-fronts are parallel, iisect the angles contained hy planes 
passing through the normal to the wave-front and the vectors (optic 
axes) A'j fx'. 

419.] As in § 409, the normal velocity is given by 
v^ ^SsTCJysr = 2SX.''aSf/tsy-p'^^ 

= / + ; 


[This transformation, effected by means of the value of or in 
§ 417, is left to the reader.] 

HencCj if w^, v^ be the velocities of the two waves whose normal 
is a, „2 _ ^.| ^ 2 T. r\'a r/a 

oc sin K'a sin ju'o. 
That is, the difference of the squares of the velocities of the two waves 
varies as the product of the sines of the angles between the normal to 
the wave-front and the optic axes (A', \j.'). 
420.] We have, obviously. 

Hence v^=p'^^ {T± S). VK'a Ff/a. 

The equation of the index surface, for which 

Tp = -, Up = a, 


is therefore 1 = -p'p^ + {T±S). Fx'p Fpfp. 

This will, of course, become the equation of the reciprocal of the 

index-surface, i.e. the wave-surface, if we put for the function ^ its 

reciprocal : i. e. if in the values of A', p.', p' we put - , y- , - for 

a, b, c respectively. We have then, and indeed it might have been 
deduced even more simply as a transformation of § 409 (7), 

\ = -pp^i;-{T±S).F\pFp.p, 
as another form of the equation of Fresnel's wave. 


If we employ the i, k transformation of § 1 2 1, this may be written, 
as the student may easily prove, in the form 

421. J We may now, in furtherance of our object, which is to 
give varied examples of quaternions, not complete treatment of any 
one subject, proceed to deduce some of the properties of the wave- 
surface from the diflFerent forms of its equation which we have 

422.] Fresnel's construction of the wave hy points. 

From § 273 (4) we see at once that the lengths of the principal 
semidiameters of the central section of the ellipsoid 

Sp<^-^p = 1, 
by the plane Sap = 0, 

are determined by the equation 

If these lengths be laid off along a, the central perpendicular to the 
cutting plane, their extremities lie on a surface for which a = Vp^ 
and Tp has values determined by the equation. 

Hence the equation of the locus is 

as in §§409, 414. ^P (r^-P'^V = 0, 

Of course the index-surface is derived from the reciprocal ellip- 
soid Sp>^p = 1 
by the same construction. 

423.] Again, in the equation 

suppose VKp = 0, or F/xp = 0, 

we obviously have 

U\ , Up. 

P = ±—7= or p = ±—=> 
vj) vp 

and there are therefore four singular points. 

To find the nature of the surface near these points put 

P = V^ + ^' 
where Tsr is very smallj and reject terms above the first order in 
Ttsr. The equation of the wave becomes, in the neighbourhood of 
the singular point, 

2^35^^ + /S.OT r. X VXp. = ±T. TAot FX/x, 
which belongs to a cone of the second order. 

424.] From the similarity of its equation to that of the wave, it 

248 QUATERNIONS. [425. 

is obvious that the index-surface also has four conical cusps. As 
an infinite number of tangent planes can be drawn at such a point, 
the reciprocal surface must be capable of being touched by a plane 
at an infinite number of points ; so that the wave-surface has four 
tangent planes which touch it along ridges. 

To find their form, let us employ the last form of equation of the 
wave in § 420. If we put 

Trip=TrKp, (1) 

we have the equation of a cone of the second degree. It meets the 
wave at its intersections with the planes 

S{l-K)p=+{K^-i^) (2) 

Now the wave-surface is touched by these planes, because we cannot 
have the quantity on the first side of this equation greater in abso- 
lute magnitude than that on the second, so long as p satisfies the 
equation of the wave. 

That the curves of contact are circles appears at once firom (1) 
and (2), for they give in combination 

p2 = +5(t + K)p, (3) 

the equations of two spheres on which the curves in question are 

The diameter of this circular ridge is 

[Simple as these processes are, the student will find on trial that 
the equation Sp{<f>~''-—p~^)~'^p = 0, 

gives the results quite as simply. For we have only to examine 
the eases in which — p"^ has the value of one of the roots of the 
symbolical cubic in (^"^. In the present case Tp = b is the only one 
which requires to be studied.] 

425.] By § 41 3, we see that the auxiliary vector of the succeed- 
ing section, viz. 

is parallel to the direction of the force of restitution, 0in-. Hence, 
as Hamilton has shewn, the equation of the wave, in the form 

Srp = 0, 
(4) of §414, indicates that fJie direction of the force of restitution is 
perpendicular to the ray. 

Again, as for any one versor of a vector of the wave there are two 
values of the tensor, which are found from the equation 


we see by § 422 that the lines of vibration for a given plane front 
are parallel to the axes of any section of the ellipsoid, 

S.p(t>-^p = 1 
made hy a plane parallel to the front ; or to the tangents to the lines 
of curvature at a point where the tangent plane is parallel to the wave- 

426.] Again, a curve which is drawn on the loave-surface so as to' 
touch at each point the corresponding line of vibration has 

Hence S(ppdp = 0, or Sp^p = C, 

so that such, curves are the intersections of the wave with a series 
of ellipsoids concentric with it. 

427,] For curves cutting at right angles the lines. of vibration we 
have dp II Fp(j)-^ ((/)-! -p-^)-V 

Hence Spdp = 0, or Tp = C, 

so that the curves in question lie on concentric spheres. 
They are also spherical conies, because where 

Tp = C 
the equation of the wave becomes 

the equation of a cyclic cone, whose vertex is at the common centre 
of the sphere and the wave-surface, and which cuts them in their 
curve of intersection. (Quarterly Math. Journal, 1859.) 

428.] As another example we take the case of the action of 
electric currents on one another or on magnets; and the mutual 
action of permanent magnets. 

A comparison between the processes we employ and those of 
Ampere {Theorie des Phenomenes Mectrodynamiques, ^c, many of 
which are well given by Murphy in his Electricity) will at once 
shew how much is gained in simplicity and directness by the use of 

The same gain in simplicity will be noticed in the investigations 
of the mutual effects of permanent magnets, where the resultant 
forces and couples are at once introduced in their most natural and 
direct forms. 

429.] Ampere's experimental laws may be stated as follows : 

I. Equal and opposite currents in the same conductor produce 
equal and opposite effects on other conductors : whence it follows 

250 QUATERNIONS. [430. 

that an element of one current has no effect on an element of an- 
other which lies in the plane bisecting the former at right angles. 

II. The effect of a conductor bent or twisted in any manner is 
equivalent to that of a straight one, provided that the two are 
traversed by equal currents, and the former nearly coincides with 
the latter. 

III. No closed circuit can set in motion an element of 'a circular 
conductor about an axis through the centre of the circle and per- 
pendicular to its plane. 

IV. In similar systems traversed by equal currents the forces are 

To these we add the assumption that the action between two 
elements of currents is in the straight line joining them : and two 
others, viz. that the effect of any element of a current on another is 
directly as the product of the strengths of the currents^ and of the 
lengths of the elements. 

430.] Let there be two closed currents whose strengths are a 
and a^; let a, Oj be elements of these, a being the vector joining 
their middle points. Then the effect of a on oj must, when resolved 
along Oj, be a complete differential with respect to a (i.e. with respect 
to the three independent variables involved in a), since the total 
resolved effect of the closed circuit of which a' is an element is zero 
by III. 

Also by I, II, the effect is a function of Ta, Saa, Saa^, and 8a a^, 
since these are suflScient to resolve a and Oj into elements parallel 
and perpendicular to each other and to a. Hence the mutual effect 
is aa-JJaf{Ta, Saa, Saa^, Si/aj), 

and the resolved effect parallel to a^ is 

aiZj SUai TJaf. 
Also, that action and reaction may be equal in absolute magnitude, 
ymust be symmetrical in Sao! and Saa-^. Again, d (as differential 
of a) can enter only to the first power, and must appear in each term 

Hence f^ASaa-^-\-^SaaSaa^. 

But, by .IV, this must be independent of the dimensions of the 
system. Hence J is of — 2 and ^ of — 4 dimensions in Ta. There- 

^""■^ ^ {ASaa^Sda^ + BSaa'S^aa^} 

is a complete differential, with respect to a, if da = a. Let 


where C is a constant deperaing on the units employed, therefore 

=.-=r; baa, 

2Ta^ ~ Ta 

and the resolved effect 

Gaa^ S^aa^ Saa, „„ , , » o /o ^ 

"^ W^ IhF "^ 1 Ta Ta^ ^ ~ "i +^ 1^ 

= Caa^ „ y,^g {S. Vaa' Faa^-{-\ Saa'Saa^ . 

The factor in brackets is evidently proportional in the ordinary 
notation to sin 6 sin 6'cos ia — \ cos 6 cos 6'. 

431.] Thus the whole force is 

Caa-^a , S^aa-^ _ Caa■^^a , S'^aa' 

as we should expect, d-^a being = a^. [This may easily be trans- 
formed into 2Caa,Ua 

which is the quaternion expression for Ampere's well-known form.] 
432.] The whole effect on Oj of the closed circuit, of which a is 
an element, is therefore 

Cfeffj C a JSaa-^^ 

H f a 
J Saa^ 

2 J Saa, Ta 


between proper limits. As the integrated part is the same at both 
limits, the effect is 

^•^^IF a I, a f^°-^' fdUa 

- V-^"^^' ''^''' ^=J'T^=J-^' 

and depends on the form of the closed circuit. 

433.] This vector ^, which is of great importance in the whole 
theory of the effects of closed or indefinitely extended circuits, cor- 
responds to the line which is called by Ampere " direcfrice de V action 
electrodynamique" It has a definite value at each point of space, 
independent of the existence of any other current. 

Consider the circuit a polygon whose sides are indefinitely small; 
join its angular points with any assumed point, erect at the latter, 
perpendicular to the plane of each elementary triangle so formed, a 


vector whose length is - > where to is the vertical angle of the tri- 

252 QUATERNIONS. [434- 

angle and r the length of one of the containing sides ; the sum of 
such vectors is the " directrice" at the assumed point. 

434.] The meve/orm of the result of § 432 shews at once that 
if the element Oj he turned about its middle point, the direction of the 
resultant action is confined to the plane whose normal is j3. 

Suppose that the element Oj is forced to remain perpendicular to 
some given vector 6, we have 

Soj^b = 0, 
and the whole action in its plane of motion is proportional to 


But r.bra^li=-a^S^b. 

Hence the action is evidently constant for all possible positions 
of a^ ; or 

The effect of any system of closed currents on an element of a con- 
ductor which is restricted to a given plane is {in that plane) independent 
of the direction of the element. 

435.] Let the closed current be plane and very small. Let e 
(where Tt =■ 1 ) be its normal, and let y be the vector of any point 
within it (as the centre of inertia of its area) ; the middle point of 
oj being the origin of vectors. 

Let a = y + p; therefore a'= p, 

and .-/•^""-/• ^(y + P) / 

and P-J Ta?-J T(y + pY 


to a sufficient approximation. 

Now (between limits) fVpp'= 2Ae, 
where A is the area of the closed circuit. 
Also generally 

fVyp'Syp =^\{SypVyp^y7.yfVpp') 
= (between limits) AyVye. 
Hence for this case 

A , 3yFye>. 

^=TyS{^' + -^) 

A ( 3y% x 

- Ty^\ '^ Ty"^ )' 

436.] If, instead of one small plane closed current, there be a 
series of such, of equal area, disposed regularly in a tubular form, 
let X be the distance between two consecutive currents measured 
along the axis of the tube; then, putting y'= xs, we have for the 
whole effect of such a set of currents on a^ 


g-^^«i V. fry J. ^y^yY' \ 

CAaa.^ Va■^y ,. , t -j. x 

= — - — - „ 3 (between proper limits). 

If the axis of the tubular arrangement be a closed curve this will 
evidently vanish. Hence a closed solenoid exerts no influence on an 
element of a conductor. The same is evidently true if the solenoid he 
indefinite in both directions. 

If the axis extend to infinity in one direction, and y^ be the 
vector of the other extremity, the effect is 

CAaa^ VoiVo 

and is thevefove perpendicular to the element and to the line joining it 
with the extremity of the solenoid. It is evidently inversely as Ty'% 
and directly as the sine of the angle contained letmeen the direction of 
the element and that of the line joining the latter with the extremity of 
the solenoid. It is also inversely as x, and therefore directly as the 
number of currents in a unit of the axis of the solenoid. 

437.] To find the effect of the whole circuit whose element is Oj 
on the extremity of the solenoid, we must change the sign of the 
above and put a^ = y^; therefore the effect is 
_ CAaa^ r Vygyg 
2x J Ty% ' 
an integral of the species considered in § 432 whose value is easily 
assigned in particular cases. 

438.] Suppose the conductor to le straight, and indefinitely extended 
in both directions. 

Let ho be the vector perpendicular to it from the extremity of 
the canal, and let the conductor be || 77, where Td = Tri = 1 . 

Therefore yg = h6+yr} (where y is a scalar), 

TyoVo = A/J'rie, 
and the integral in § 436 is 


J —CD 


-00 {h-'+y^f h 
The whole effect is therefore 

and is thus perpendicular to the plane passing through the conductor 
and the extremity of the canal, and varies inversely as the distance of 
the latter from the conductor. 




This is exactly the observed effect of an indefinite straight current 
on a magnetic pole, or particle of free ina;gnetism. 

439.] Suppose the conductor to be circular, and the pole nearly in its 

Let UPD be the conductor, A£ its axis, and C the pole ; £C 
perpendicular to A£, and small in comparison with AE = h the 
radius of the circle. 

Let AJB be Oji, 

BC=hk, AP = h{jx + i 

wJ 'sm-" '•sm.-' 



CP = y =. aii-\-bk—h{jx-\-ky). 

• [Fyy 
And the effect on C<x -^fy , 

''6' {{h—by)i+a^coJ + aiyk} 
where the integral extends to the whole circuit. 

440.] Suppose in particular C to be one pole of a small magnet 
or solenoid CC whose length is 2 1, and whose middle point is at Q 
and distant a from the centre of the conductor. 
Let LGGB = A. Then evidently 

a^=- a + l cos A, 
i = ^ sin A. 
Also the effect on C becomes, i£ al + b^+h'^ = A', 

^J ^{{h-by)i^a^x3^a^h\ (l + _/ + _ —^ + ...} 

15 hHH 



"•" "a^ '^Y a* 




since for the whole circuit 

/ey +1 = 0, 

f&xy'^ = 0. 
If we suppose the centre of the magnet fixed, the vector axis of 
the couple produced by the action of the current on C is 

IV. {i cos A + ^ sin ^)j-M- 

If A, &c. be now developed in powers of I, this at once becomes 
■77^^^ sin A .C 6 a^ cos A ISa^^^cos^A 3P 

(a2 + /J2)f -^1 ~ a'^ + A^ + (a^ + A^)" '^T^ 

SlHin^A 15 A^Psin^A _ {a + lcosA)lcosA y 5 a^ cos A n | 
" a^ + A^ +T (a2+F)2~ ^^^^ y ~ a'-^-A'^ >V 

Putting —I for I and changing the sign of the whole to get that 
for pole C , we have for the vector axis of the complete couple 
4TrA2;sinA.f ^ ^2(4a2_F)(4-5 sin^A) ) 

which is almost exactly proportional to sin A if la = A and I be 

On this depends a modification of the tangent galvanometer. 
(Bravais, Ann. de CAimie, xxxviii. 309.) 

441.] As before, the effect of an indefinite solenoid on a^ is 
GAaa^^ Va-^y 

Now suppose a^ to be an element of a small plane circuit, 8 the 
vector of the centre of inertia of its area, the pole of the solenoid 
being origin. 

Let y = 8 + jO, then a^ = p. 

The whole effect is therefore 

_ CAaai f r{b + p)p' 
2« 7 T{5+pf 
_ CAA^aa^ / 38^>. 

where A^ and e^ are, for the new circuit, what A and e were for the 

Let the new circuit also belong to an indefinite solenoid, and 
let 6o be the vector joining the poles of the two solenoids. Then 
the mutual effect is 

256 QUATERNIONS. [442. 

2xx^ J ^m "^ n' ) 

_ CAA-^aa^ \ Ub^ 

- 2wx, {n^^°^{n^' 

which is exactly the mutual effect of two magnetic poles. Two finite 

solenoids, therefore, act on each other exactly as two magnets, and the 

pole of an indefinite solenoid acts as a particle of free magnetism, 

442.] The mutual attraction of two indefinitely small plane closed 

circuits, whose normals are e and e^, may evidently be deduced by 

twice diflFerentiating the expression -f=j-^ for the mutual action of 

the poles of two indefinite solenoids, making db in one differentiation 
II f and in the other || e^. 

But it may also be calculated directly by a process which will 
give us in addition the couple impressed on one of the circuits by 
the other," supposing for simplicity the first to be circular. 

Let A and B be the centres of inertia of the areas of A and B, 
« and e^ vectors normal to their planes, o- any vector radius of B, 
AB = p. 

Then whole effect on </, by §§ 432, 435, 

'Tifi + crf 

5 r^'^ I 


'V+ Tifi + .f V 

^7^i^'^Hl + l^)+ Tl3^ i^ + ^^J 

But between proper limits, 

frir'rtSdu =-A:ir,r]re€^, 
for generally fn'n 86<r = -k{ Fr,crSda- +7.7,7. QfT<T<j'). 

Hence, after a reduction or two, we find that the whole force 
exerted by A on the centre of inertia of the area of B 


This, as already observe^ may be at once found by twice differ- 
entiating m;52' ^^ ^^® same way the vector moment, due to A, 
about the centre of inertia of £, 

These expressions for the whole force of one small magnet on the 
centre of inertia of another, and the couple about the latter, seem 
to be the simplest that can be given. It is easy to deduce from 
them the ordinary forms. For instance, the whole resultant couple 
on the second magnet 

oc ^ 


may easily be shewn to coincide with that given by Ellis {Camh. 
Math. Journal, iv. 95), though it seems to lose in simplicity and 
capability of interpretation by such modifications. 

443.] The above formulae shew that the whole force exerted by 
one small magnet M, on the centre of inertia of another m, consists 
of four terms which are, in order, 

1st. In the line joining the magnets, and proportional to the cosine 
of their mutual inclination. 

2nd. In the same line, and proportional to five times the product of 
the cosines of their respective inclinations to this line. 

3rd and 4th. Parallel to { ]■ and proportional to the cosine of the 

M .... 

inclination o/" { ^ to the joining line. 

All these forces are, in addition, inversely as the fourth power of 
the distance between the magnets. 

For the couples about the centre of inertia of m we have 

1st. A couple whose axis is perpendicular to each magnet, and which 
is as the sine of their mutual inclination. 

.2nd. A couple whose axis is perpendicular to m and to the line 
joining the magnets, and whose moment is as three times the product of 
the sine of the inclination qfm, and the cosine of the inclination o^M, 
to the joining line. 

In addition these couples vary inversely as the third power of the 
distance between the magnets. 

258 QUATEENIONS. [444. 

[These results afford a good example of what has been called the 
internal nature of the methods of quaternions, reducing, as they do 
at once, the forces and couples to others independent of any lines of 
reference, other than those necessarily belonging to the system 
under consideration. To shew their ready applicability, let us take 
a Theorem due to Gauss.] 

444.] If two small magnets he at right angles to each other, the 
moment of rotation of the first is anproximately twice as great when the 
axis of the second passes through the centre of the first, as when the 
axis of the first passes through the centre of the second. 

In the first case e |{ y3 J.ej^ ; 

C 2 C" 

therefore moment = ^T(efi-3€€i) = ^yeej. 

In the second eil|/3±e; 

therefore moment = -=j-^Tee-^. Hence the theorem. 

445.] Again, we may easily reproduce the results of § 442, if for 
the two small circuits we suppose two small mag^nets perpendicular 
to their planes to be substituted. (3 is then the vector joining the 
middle points of these magnets, and by changing the tensors we 
may take 2e and 2ej^ as the vector lengths of the magnets. 

Hence evidently the mutual effect 

which is easily reducible to 

as before, if smaller terms be omitted. 

If we operate with V. e^ on the two first terms of the unreduced 
expression, and take the difference between this result and the same 
with the sign of e^ changed, we have the whole vector axis of the 
couple on the magnet 2ei, which is therefore, as- before, seen to be 
proportional to 

446.] We might apply the foregoing formulae with great ease 
to other cases treated by Ampere, De Montferrand, &c. — or to two 
finite circular conductors as in Weber's Dynamometer — but in 
general the only difficulty is in the integration, which even in some 
of the simplest cases involves elliptic functions, &c., &c. {Quarterly 
Math. Journal, 1860.) 


447.] Let F{y) be the potential of any system upon a unit 
particle at the extremity of y. 

F{y) = c (1) 

is the equation of a level surface. 

Let the differential of (1) be 

Svdy=0, (2) 

then v is a vector normal to (1), and is therefore the direction of the 

But, passing to a proximate level surface, we have Svby = bC. ' 

Make by=xv, then —a;Tv^ = bC, 

Hence v expresses the force in magnitude also. (§ 368.) 
Now by § 435 we have for the vector force exerted by a small 
plane closed circuit on a particle of free magnetism the expression 

A , ZySye\ 

omitting the factors depending on the strength of the current and 
the strength of magnetism of the particle. 
Hence the potential, by (2) and (1), 

oc ' - 


area of circuit projected perpendicular to y 

oc spherical opening subtended by circuit. 
The constant is omitted in the integration, as the potential must 
evidently vanish for infinite values of Ty. 

By means of Ampere's idea of breaking up a finite circuit into 
an indefinite number of indefinitely small ones, it is evident that 
the above result may be at once ex- 
tended to the case of such a finite closed 

448.] Quaternions give a simple me- 
thod of deducing the well-known pro- 
perty of the Magnetic Curves. 

Let A, A be two equal magnetic 
poles, whose vector distance, 2 a, is bi- 
sected in 0, QQ' an indefinitely small 

magnet whose length is Ip , where p-= OP^ Then evidently, taking 

S 2 

260 QUATEENIONS. [449. 

r{p+a)p' _ r{p-a)p' 

T{p + af - ± T{p-aY' 

where the upper or lower sign is to be taken according as the poles 

are like or unlike. 

Operate by S. Vap, 

Sap{p + af—Sa{p + a)Sp'{p-ira) ^ , -.t, i 
^^^ y(p^„)S = ± {s^°ie With -a], 


or S.af {-^-^V{p-\-a)-= + {same with —a\, 
^p + a^ 

i.e. SadU(p + a) = + SadU{p — a), 

Sa { U{p + a) + U{p—a)} = const., 

or cos Z OAP ± cos / OA'P = const., 

the property referred to. 

If the poles be unequal, one of the terms to the left must be 

multiplied by the ratio of their strengths. 

4(49.] K the vector of any point be denoted by 

p = ix+Jt/ + iz, (1) 

there are many physically interesting and important transformations 

depending upon the effects of the quaternion operator 

„ ■ d . d , d ,. 

^ = ^^+^^ + ^^^ (') 

on various functions of p. When the function of p is a scalar, the 

effect of V is to give the vector of most rapid increase. Its effect 

on a vector function is indicated briefly in § 369. 

450.] We commence with one or two simple examples, which 

are not only interesting, but very useful in transformations. 

7 * 

V/) = fiy- +&c.)(«« + &c.) =— 3, (3) 

ViTpf = n{TpY-'^VTp = n{TpY-^p; (5) 

and, of course, v^-^ = -^^^; (5)i 

Tp Tp^ ~ Tp"^ 

whence, V ^^ =- ^rj =- ^rr. (6) 

and, of course, V2y- = — VyY= (6)^ 

Also, Vp =-3 = TpVUp + VTp.Up = TpVUp-l, 

■'■ ^^P = -T^ (7) 


451.] By the help of the above results, of which (6) is especially 
useful (though obvious on other grounds), and (4) and (7) very 
remarkable, we may easily find the effect of V upon more complex 

Thus, VSap=-V{aic-{-kc.) = -a, (1) 

Vrap = — VFpa =—V{pa—Sap) = 3a— a= 2a (2) 


^ ^ap _ 2a ZpVap _ 2ap^ + 3pF'ap _ ap^ — 3pSap . . 

T^^Tp^~'T^~ Tf^ " Tp^ ^' 


„ Vap p^ Sahp — ZSapSphp Sahp ZSapSpbp » '^"P / \ 

'^•8PV^= jp =_______ = _6_.(4) 

This is a very useful transformation in various physical applica- 
tions. By (6) it can be put in the sometimes more convenient form 

S.hpV^=hS.aVy~ (5) 

And it is worthy of remark that, as may easily be seen, —S may be 
put for V in the left-hand member of the equation. 
452.] We have also 

'f7r.0py=V{^Syp-pSPy + ySpp] =-yfi + 3S^y-l3y ^SjSy. (1) 
Hence, if <j) be any linear and vector function of the form 

(j)p = a + ^F.fipy + mp, (2) 

i.e. a self-conjugate function with a constant vector added, then 

V(f>p = 2S^y—3m = scalar (3) 

Hence, an integral of 

Vo- = scalar constant, is <t = (l>p (4) 

If the constant value of Vo- contain a vector part, there will be 
terms of the form Fep in the expression for a; which will then ex- 
press a- distortion accompanied by rotation. (§371.) 

Also, a solution of V^" = « (where q and a are quaternions) is 

q = SCp+Ffp + (Pp. 
It may be remarked also, as of considerable importance in phy- 
sical applications, that, by (1) and (2) of § 451, 

V{S+ir)ap = 0, 
but we cannot here enter into details on this point. 

453.] It would be easy to give many more of these transforma- 
tions, which really present no difiiculty ; but it is sufficient to shew 

262 QUATERNIONS. [454. 

the, ready applicability to physical questions of one or two of those 
already obtained ; a property of great importance, as extensions of 
mathematical physics are far more valuable than mere analytical or 
geometrical theorems. 

Thus, if (7 be the vector-displacement of that point of a homo- 
geneous elastic solid whose vector is p, we have, j» being the con- 
sequent pressure producedj 

Vj9-)-W = 0, (1) 

whence <S'SpV^<j-= —SbpVp = 8jb, a complete differential (2) 

Also, generally, p = kSVa, 

and if the solid be incompressible 

S^cT= (3) 

Thomson has shewn {Caml. and Bub. Math. Journal, ii. p. 62), 
that the forces produced by given distributions of matter, electricity, 
magnetism, or galvanic currents, can be represented at every point 
by displacements of such a solid producible by external forces. It 
may be useful to give his analysis, with some additions, in a qua- 
ternion form, to shew the insight gained by the simplicity of the 
present method. 

454. j Thus, if Scrbp = 8 =,- , we may write each equal to 


This gives (T = —Vyj^, 


the vector-force exerted by one particle of matter or free electricity 

on another. This value of o- evidently satisfies (2) and (3). 

Again, if S.hpVa = 6 j—g , either is equal to 

-8.hpV^ by (4) of §451. 

Here a particular case is 


which is the vector-force exerted by an element a of a current upon 
a particle of magnetism at p. (§ 436.) 
455.] Also, by §451 (3), 

Vap _ ap^ — ZpSap 


and we see by §§ 435, 436 that this is the vector-force exerted by a 
small plane current at the origin (its plane being perpendicular to a) 
upon a magnetic particle, or pole of a solenoid, at p. This expres- 
sion, being a pure vector, denotes aii elementary rotation caused by 
the distortion of the solid, and it is evident that the above value of 
(T satisfies the equations (2), (3), and the distortion is therefore pro- 
ducible by external forces. Thus the effect of an element of a 
current on a magnetic particle is expressed directly by the displace- 
ment, while that of a small closed current or magnet is represented 
by the vector-axis of the rotation caused by the displacement. 

456.] Again, let ^5pVV=8^. 

It is evident that a- satisfies (2), and that the right-hand side of the 
above equation may be written 


Hence a particular case is 


and this satisfies (3) also. 

Hence the corresponding displacement is producible by external 
forces, and Vo- is the rotation axis of the element at p, and is seen 
as before to represent the vector-force exerted on a particle of mag- 
netism at p by an element a of a current at the origin. 

457.] It is interesting to observe that a particular value of o- in 
this case is ^ 

(T — —\VSaUp—yjr' 

as may easily be proved by substitution. 
Again, if Sbpa- = — 8 ~^ > 

we have evidently o- = V -jfj • 

Now, as yj^ is the potential of a. small magnet a, at the origin, 

on a particle of free magnetism at p, o- is the resultant magnetic 
force, and represents also a possible distortion of the elastic solid 
by external forces, since Vo- = V^o- = 0, and thus (2) and (3) are 
both satisfied. 

458.] We conclude with some examples of quaternion integra- 
tion of the kinds specially required for many important physical 

264 QUATERNIONS. ^ [459. 

It may perhaps be useful to commence with a different form 
of definition of the operator V, as we shall thus, if we desire it, 
entirely avoid the use of ordinary Cartesian coordinates. For this 
purpose we write 

where a is any unit-vector, the meaning of the right-hand opei'ator 
(neglecting its sign) being the rate of change of the function to which 
it is applied per unit of length in the direction of the unit- vector a. 
If a be not a unit-vector we may treat it as a vector-velocity, and 
then the right-hand operator means the- m^e of change per unit of 
time due to the change of position. 

. Let a, /3, y be any rectangular system of unit-vectors, then by a 
fundamental quaternion transformation 

V = — aSaV — ySiS/SV — ySyV = ad^ + ^d^ ^ ydy , 
which is identical with Hamilton's form so often given above. 
(Lectures, § 620.) 

459.] This mode of viewing the subject enables us to see at once 
that the effect of applying V to any scalar function of the position 
of a point is to give its vector of most rapid increase. Hence, when 
it is applied to a potential u, we have 

Vu = vector-force at p. 

It u be a velocity-potential, we obtain -the velocity of the fluid 
element at p ; and if w be the temperature of a conducting solid we 
obtain the flux of heat. Finally, whatever series of surfaces is repre- 
sented by u = C, 

the vector Vu is the normal at the point p, and its length is inversely 
as the normal distance at that point between two consecutive sur- 
faces of the series. 

Hence it is evident that 

S.dpVu =—du, 
or, as it may be written, 

—S.dpV= d; 

the left-hand member therefore expresses total differentiation in 
virtue of any arbitrary, but small, displacement dp. 

460.] To interpret the operator V.aV let us apply it to a poten- 
tial function u. Then we easily see that u may be taken under 
the vector sign, and the expression 

F{aV)u = Y.aSJu 
denotes the vector, couple due to the force at p about a point whose 
relative vector is o. 


Again, if o- be any vector function of p, we have by ordinary 
quaternion operations 

r(aV).(r = S.arT7(T-\.a£Vc7 — VSa<T. 
The meaning of the third term (in which it is of course understood 
that V operates on n- alone) is obvious from what precedes. It 
remains that we explain the other terms. 

461.] These involve the very important quantities (not operators 
such as the expressions we have been hitherto considering), 

S.V(T and V.V<t, 
which form the basis of our investigations. Let us look upon <t as 
the displacement, or as the velocity, of a point situated at p, and 
consider the group of points situated near to that at p, as the quan- 
tities to be interpreted have reference to the deformation of the 

462.] Let T be the vector of one of the group relative to that 
situated at p. Then after a small interval of time t, the actual 
coordinates become p + i^c 

and p + r+t{(T—8{TV)a) 

by the definition of V in § 458. Hence, if be the linear and vector 
function representing the deformation of the group, we have 

^r = T—tS{TV)<T. 

The farther solution is rendered veiy simple by the fact that we 
may assume t to be so small that its square and higher powers 
may be neglected. 

If <^' be the function conjugate to <^, we have 

^'t = T—tVST<T. 

Hence <^r = i(<^ + (^')r + i(0 — c^')'' 

= t--[-s(tv)o-+ v-Sro-]— ^ r.Trv(T. 

The first three terms form a self-conjugate linear and vector func- 
tion of r, which we may denote for a moment by utt. Hence 

(j)T = ■^r—rf'.rVVa; 

or, omitting f^ as above. 

Hence the deformation may be decomposed into — (1) the pure strain 
■ST, (2) the rotation t „ 

Thus the vector-axis of rotation of the group is 

266 QUATERNIONS. [463. 

If we were content to avail ourselves of the ordinary results of 
Cartesian investigations, we might at once have reached this con- 
clusion by noticing that 

v% dzJ •'\dz Ax' \dx df 
and remembering as in (§ 362) the formulae of Stokes and Helmholtz. 
463.] In the same way, as 

SV<T=—— — — — — 

dx dy dz^ 

we recognise the cubical compression of the group of points considered. 

It would be easy to give this a more strictly quaternionic form by 

employing the definition of § 458. Butj working with quaternions, 

we ought to obtain all our results by their help alone ; so that we 

proceed to prove the above result by finding the volume of the 

ellipsoid into which an originally spherical group of points has been 

distorted in time t. 

For this purpose, we refer again to the equation of deformation 

and form the cubic in ^ according to Hamilton's exquisite process. 
We easily obtaiuj remembering that <^ is to be neglected*, 
(i = ^^-{% — tSV(i)<^^ + {^ — nSV<j)^—{\—tSV<T), 
or = (^-1)2(^—1 + i!5Vcr). 

The roots of this equation are the ratios of the diameters of the 
ellipsoid whose directions are unchanged to that of the sphere. 
Hence the volume is increased by the factor 

1— i!5Vo-, 
from which the truth of the preceding statement is manifest. 

* Thus, in Hamilton's notation, X, ;*, v being any three non-coplanar vectors, and 
m, m, , «i2 ^^ coefficients of the cubic, 
— ttSXnv = S-ip'f^^'iup'v 

^S.(\-tVSKa)(yiiv-tVii^8va + tVy'78ii<T) 
=8.\iiv-t[S.iivV8\<T + 8.v\'78n<r + 8.\ii'VSva'i 
= S.\iiv-t8. l\8.iivV + ii8.v\V + v8.\iiV'\ a 
'miS.\iiv=S.\(t>'ii<l)'v + 8.fi^'v<l>'k + 8^vip'\(p'fi 

=8.K (ynv-tVnV8vtr+ tVvV8na) + &c. 
= 38.\nv-2t8Va8.\iiv. 
—m^S .\nv = 8 .\ii<p'v + 8. iiv^'\ + 8.v\(j> n 
=8.\iJi.v—t8.K/iV8va + &c. 
= S8.\iiv—t8V<r8.\iiv 


464.] As the process in. last section depends essentially on the 
use of a non-conjugate vector function, with which the reader is less 
likely to be acquainted than with the more usually employed forms, 
I add another investigation. 

Let ■BT = ^T = T—tS{TV)(r. 

Then t = (f - V = t:j + tS (in- V) a. 

Hence since if, before distortion, the group formed a sphere of radius 

1, we have Tt = 1, 

the equation of the ellipsoid is 

T{'!!T + tS(:!!TV)<T)= 1, 

or ■!!r^ + 2iS-nTVS^a- = — 1. 

This may be written 

S.wx^ = S.w {nr + i VSi!7(T + tS (in- V) <t) — — I, 
where x is now self- conjugate. 

Hamilton has shewn that the reciprocal of the product of the 
squares of the semiaxes is 

— 'S'-XWX'^. 
whatever rectangular system of unit-vectors is denoted by i, j, h. 
Substituting the value of x, we have 

—8.{i^tVSi(T^t8{iV)a) (y + &c.) (/^-|-&c.) 
= —S.{i■^r tVSia + tS («V) a){i+2 tiSVa— iS{iV)a- tVSicr) 
^ l+2tSVa. 
The ratio of volumes of the ellipsoid and sphere is therefore, as 

before, 1 

, = 1 - fSVcr. 

VI + 2tSV(T 

465.] In what follows we have constantly to deal with integrals 
extended over a closed surface, compared with others taken through 
the space enclosed by such a surface ; or with integrals over a 
limited surface, compared with others taken round its bounding 
curve. The notation employed is as foUows. If Q, per unit of 
length, of surface, or of volume, at the point p, Q being any qua- 
ternion, be the quantity to be summed, these sums will be denoted 
by f/qds and Jf/qds, 

when comparing integrals over a closed surface with others through 
the enclosed space ; and by 

f/qds and /QTdp, 
when comparing integrals over an unclosed surface with others round 
its boundary. No ambiguity is likely to arise from the double use of 

268 QUATERNIONS. [466. 

for its meaning in any case will be obvious from the integral with 
which it is compared. 

466.J We have just shewn thatj if a- be the vector displacement 
of a point originally situated at 

p = ix+jy + kz, 
then S.Va- 

expresses the increase of density of aggregation of the points of the 
system caused by the displacement. 

467.] Suppose, now, space to be uniformly filled with points, and 
a closed surface S to be drawn, through which the points can freely 
move when displaced. 

Then it is clear that the increase of number of points within the 
space 2, caused by a displacement, may be obtained by either of two 
processes — by taking account of the increase of density at all points 
within 2, or by estimating the excess of those which pass inwards 
through the surface over those which pass outwards. These are 
the principles usually employed (for a mere element of volume) in 
forming the so-called ' Equation of Continuity.' 

Let V be the normal to 2 at the point p, drawn outwards, then 
we have at once (by equating the two different expressions of the 
same quantity above explained) the equation 
///S.Vads =//S.<rUvds, 

which is our fundamental equation so long as we deal with triple 

468.] As a first and very simple example of its use, suppose o- 
to represent the vector force exerted upon a unit particle at p (of 
ordinary matter, electricity, or magnetism) by any distribution of 
attracting matter, electricityj or magnetism partly outsidcj partly 
inside 2. Then, if P be the potential at p, 

<r = VP, 
and if r be the density of the attracting matter, &c., at p, 

V(T=V^P = 4irr 
by Poisson's extension of Laplace's equation. 

Substituting in the fundamental equation, we have 
4:i:///rds= 4:-nM=//S.VPUvds, 

where M denotes the whole quantity of matter, &c., inside 2. This 
is a well-known theorem. 

469.] Let P and Pj be any scalar functions of p, we can of course 
find the distribution of matter, &c., requisite to make either of them 


the potential at p ; for, if fhe necessary densities be r and i\ re- 
spectively, we have as before 

Now V (P VPi) = VP VPj + P V^Pi , 

Hence, if in the above formula we put 

we obtain 

J/fS.VPVPJs = -///PV^P,ds+//PS.VP,Uvds, 
= -///P^^'Pds +//P,S.VP Uvds, 
which are the common forms of Greenes Theorem. Sir W. Thomson's 
extension of it follows at once from the same proof. 

470.] If Pj be a many- valued function, but VPj single- valued, 
and if 2 be a multiply-connected* space, the above expressions 
require a modification which was first shewn to be necessary by 
Helmholtz, and first supplied by Thomson. For simplicity, suppose 
2 to be doubly-connected (as a ring or endless rod, whether knotted 
or not). Then if it be cut through by a surface s, it will become 
simply-connected, but the surface-integrals have to be increased by 
terms depending upon the portions thus added to the whole surface. 
In the first form of Greenes Theorem, just given, the only term 
altered is the last : and it is obvious that if jo^ be the increase of P^ 
after a complete circuit of the ring, the portion to be added to the 
right-hand side of the equation is 

taken over the cutting surface only. Similar modifications are 
easily seen to be produced by each additional complexity in the 
space 2. 

471.] The immediate consequences of Green's theorem are well 
' known, so that I take only one instance. 

Let P and P^ be the potentials of one and the same distribution 
of matter, and let none of it be within 2. Then we have 

///{vpyds =f/ps.vpuvds, 

so that if VP is zero all over the surface of 2, it is zero all through 
the interior, i.e., the potential is constant inside 2. If P be the 
velocity-potential in the irrotational motion of an incompressible 
fluid, this equation shews that there can be no such motion of the 

* Called by Helmholtz, after Eiemann, mehrfach zusammenhdngend. In translating 
Helmholtz'a paper {Phil. Mag. 1867) I used the above as an English equivalent. Sir 
■W.Thomson in his great paper on Yortex Motion {Trans. B. S-.E. 1868) uses the ex- 
pression "multiply-continuous." 

270 QUATERNIONS. [472. 

fluid unlesB there is a normal motion at some part of the bounding 
surface, so long at least as 2 is simply-connected. 
Again, if 2 is an equipotential surface, 

f/f(ypfd, = Pf/s.vPUvds = Pf//v^Pds 

by the fundamental theorem. But there is by hypothesis no matter 
inside 2, so this shews that the potential is constant throughout 
the interior. Thus there can be no equipotential surface, not in- 
cluding some of the attracting matter, within which the potential 
can change. Thus it cannot have a maximum or minimum value 
at points unoccupied by matter. 

472.] If, in the fundamental theorem, we suppose 

a- =Vt, 
which imposes the condition that 

S.V(T = 0, 
i.e., that the <r displacement is effected without condensation, it 
becomes //S.VrUvds =///S.V^Tds = 0. 

Suppose any closed curve to be traced on the surface 2, dividing 
it into two parts. This equation shews that the surface-integral is 
the same for both parts, the difference of sign being due to the fact 
that the normal is drawn in opposite directions on the two parts. 
Hence we see that, with the above limitation of the value of a, the 
double integral is the same for all surfaces bounded by a given 
closed curve. It must therefore be expressible by a single integral 
taken round the cui-ve. The value of this integral will presently 
be determined. 

473.] The theorem of § 467 may be written 

///V^Pds =//S.UvVPds =//S{UvV)Pds. 
From this we conclude at once that if 

^ = iP+JP^ + kP^, 
(which may, of course, represent any vector whatever) we have 

or, if V^o- = T, 


This gives us the means of representing, by a surface-integral, a 
vector-integral taken through a definite space. We have already 
seen how to do the same for a scalar-integral — so that we can now 
express in this way, subject, however, to an ambiguity presently 
to be mentioned, the general integral 


where q is any quaternion \^atever. It is evident that it is only 
in certain classes of cases that we can exnect a perfectly definite 
expression of such a volume-integral in terms of a surface-integral. 
474.] In the above formula for a vector-integral there may 
present itself an ambiguity introduced by the inverse operation 

to which we must devote a few words. The assumption 

is tantamount to saying that, as the constituents of a- are the 
potentials of certain distributions of matter, &c., those of t are the 
corresponding densities each multiplied by 4 tt. 

If, therefore, r be given throughout the space enclosed by S, 
o- is given by this equation so far only as it depends upon the 
distribution within S, and must be completed by an arbitrary vector 
depending on three potentials of mutually independent distributions 
exterior to 2. 

But, if o- be given, t is perfectly definite ; and as 
Vo- = V-^Tj 
the value of V""^ is also completely defined. These remarks must 
be carefully attended to in using the theorem above : since they 
involve as particular cases of their application many curious theorems 
in Fluid Motion^ &c. 

475.] As a particular case, the equation 
of course gives V a- := u, a scalar. 

Now, if V be the potential of a distribution whose density is u, we 
have V'^v = 4Tr?<. 

We know that this equation gives one, and but one, definite value 
for V, so that there is no ambiguity in 

V = 4tV~^?<, 

and therefore o- = — - V« is also determinate. 


476.] This shews the nature of the arbitrary term which must 

be introduced into the solution of the equation 


To solve this equation is (§ 462) to find the displacement of any 

one of a group of points when the consequent rotation is given. 

Here -SVr = -S. V FVo- = 5 W = ; 

so that, omitting the arbitrary term (§ 475), we have 


and each constituent of o- isj as above, determinate. 

272 QUATERNIONS. [477. 

Thomson * has put the solution in a form which may be written 

if we understand by y*( ) dp integrating the term in da; as if y 
and z were constants, &c. Bearing this in mind, we have as 

rv<r = i2ri[vTi+fr^dp^ 

= i{3T+/dpSVr}=T. 
477.] We now come to relations between the results of integra- 
tion extended over a non-closed surface and round its boundary. 

Let IT be any vector function of the position of a point. The 
line-integral whose value we seek as a fundamental theorem is 

where t is the vector of any point in a small closed curve, drawn 
from, a point within it, and in its plane. 

Let o-Q be the value of a- at the origin of t, then 
a- = <rf,-S(TV)crQ, 
so that /S.o-dr =z/S.(a-o-SiTV)<io)dr. 

But fdr = 0, 

because the curve is closed ; and (Tait on Mectro-Di/namics, § 1 3, 
Quarterly Math. Journal, Jan. 1860) we have generally 

fS.TVS.Oadr = \S.V{TScr^T-<Tjr.TdT). 
Here the integrated part vanishes for a closed circuit, and 

\fT.TdT = dsUv, 
where ds is the area of the small closed curve, and Uv is a unit- 
vector perpendicular to its plane. Hence 

fS.cTf^dT = S.V(TgUv.ds. 
Now, any finite portion of a surface may be broken up into small 
elements such as we have just treated, and the sign only of the 
integral along each portion of a bounding curve is changed when 
we go round it in the opposite direction. Hence, just as Ampere 
did with electric currents, substituting for a finite closed circuit 
a network of an infinite number of infinitely small ones, in each 
contiguous pair of which the common boundary is described by 
equal currents in opposite directions, we have for a finite unclosed 
surface /S.adp = jyS.Vcrllv.ds. 

There is no diflSculty in extending this result to cases in which the 
* Electrostatics and Magnetism, § 521, or Phil. Trans., 1852. 


bounding curve consists offletached ovals, or possesses multiple 
points. This theorem seems to have beeu first given by Stokes 
(Smith's Prize Esoam. 1854), in the form 


It solves the problem suggested by the result of § 472 above. 

478.] If a- represent the vector force acting on a particle of 
matter at p, —S.adp represents the work done while the particle is 
displaced along dp, so that the single integral 

of last section, taken with a negative sign, represents the work 
done during a complete cycle. When this integral vanishes it is 
evident that, if the path be divided into any two parts, the work 
spent during the particle's motiou through one part is equal to that 
gained in the other. Hence the system of forces must be con- 
servative, i. e., must do the same amount of work for all paths 
having the same extremities. 

But the equivalent double integral must also vanish. Hence a 
conservative system is such that 

//dsS.V<TUv = 0, 
whatever be the form of the finite portion of surface of which ds is 
an element. Hence, as Vo- has a fixed value at each point of space, 
while Uv may be altered at will, we must have 

rvo- = 0, 

or Vo- = scalar. 

If we call X, T, Z the component forces parallel to rectangular 
axes, this extremely simple equation is equivalent to the well-known 

dX_dY_Q ^_^^o ^_^=o 
Hy dx " ' dz dy ^ ' dx dz 

Returning to the quaternion form, as far less complex^ we see that 
Vo- = scalar = 4Trr, suppose, 
implies that o- = VP, 

where P is a scalar such that 

V2P= ^-nr; 
that is, P is the potential of a distribution of matter, magnetism, or 
statical electricity, of volume-density /. 

274 QUATERNIONS. [479. 

Hence, for a non-closed path, under conservative forces 
-fS.a-dp = -fS.VPdp 
= -/S{dpV)P 
= /da,P=/dP 

= Pi-Po, 
depending solely on the values of P at the extremities of the path. 

479.] A vector theorem, which is of great use, and which cor- 
responds to the Scalar theorem of § 473, may easily be obtained. 
Thus, with the notation already employed, 

/V.adr =/r{<T,-S(TV)<r,)dr, 

Now r{F.vr.TdT)(ra=-S{TS7)r.<TgdT-S{dTV)FT(ro, 

and d{S(,TV)r<ToT) = S{TV)r.<T^dT-\-S{dTV)ro^T. 

Subtracting, and omitting the term which is the same at both 

limits, we have fV, adr = — ¥.(¥. UvV) cr^ ds. 

Extended as above to any closed curve, this takes at once the form 

/r.(Tdp= -//ds r. ( r. Uvv) <t. 

Of course, in many cases of the attempted representation of a 
quaternion surface-integral by another taken round its bounding 
curve, we are met by ambiguities as in the case of the space- 
integral, § 474 : but their origin, both analytically and physically, 
is in general obvious. 

480.] If P be any scalar function of p, we have (by the process 
of § 477, above) 

/Pdr =/{P,-S{rV)P,)dT 

= -/S.TVPo.dT. 

But r.W.rdT = drS.TV—TS.dTV, 

and dirSrV) = drS.TV + TS.drV. 

These give 

/Pdr = -^ {TSTV-F.FTdTV)Po = dsF.UvVPg, 
Hence, for a closed curve of any form, we have 


from which the theorems of §§ 477, 479 may easily be deduced. 
481.] Commencing afresh with the fundamental integral 
put a = UjS, 

and we have ///S^Vuds =//uS.^U'vds; 


from which at once ///Vuas = f/uUvds, (1) 

or //fVTd,=/fUv.Tds (2) 

Putting WjT for r, and taking the scalar, we have 
f/f{SrVUi + u.^SVT)ds = f/n^Sr Uvds, 

whence ///(S{rV)(r + (T8.VT)ds = //(rSrUvds (3) 

483.] As one example of the important results derived from these 
simple formulae, take the following, viz. : — 

ffr.{Y<j'Uv)Tds = /f<TSTUvds-//UvS<TTds, 
where by (3) and (1) we see that the right-hand member may be 
written = //f{8{rV)(7 + <rSVT-V S(TT)ds 

= -fffr.nv<j)Tds (4) 

This, and similar formulae^ are easily applied to find the potential 
and vector-force due to various distributions of magnetism. To 
shew how this is introduced, we briefly sketch the mode of expressing 
the potential of a distribution. 

483.] Let or be the vector expressing the direction and intensity 
of magnetisation, per unit of volume, at the element ds. Then if 
the magnet be placed in a field of magnetic force whose potential 
is u, we have for its potential energy 
E = -ff/ScrVuds 

= ///uSV(rds-//tiS(rUvds. 

This shews at once that the magnetism may be resolved into a 
volume-density <S(V<7), and a surface-density —ScrUv. Hence, for a 
solenoidal distribution, S.'V(r = 0. 

What Thomson has called a lamellar distribution (PMl. Trans. 
1852), obviously requires that 

be integrable without a factor ; i. e., that 

FVa- = 0. 
A complex lamellar distribution requires that the same expression 
be integrable by the aid of a factor. If this be u, we have at once 

FV[ua) = 0, 
or S.<tV(t=0. 

With these preliminaries we see at once that (4) may be written 


Now, if T = V(-), 

where r is the distance between any external point and the element 

276 QUATERNIONS. [484- 

ds, the last term on the right is the vector-force exerted by the 
magnet on a unit-pole placed at the point. The second term on 
the right vanishes by Laplace's equation, and the first vanishes as 
above if. the distribution of magnetism be lamellar, thus giving 
Thomson's result in the form of a surface integral. 

484.] An application may be made of similar transformations to 
Ampere's Directrice de V action electrodynamique, which, § 432 above, 
is the vector-integral C^pdp 


where dp is an element of a closed circuit, and the integration 
extends round the circuit. This may be written 



so that its value as a surface integral is 

jjs {UvV)V-ds -JJuvV^ i ds. 

Of this the last term vanishes, unless the origin is in, or infinitely 

near to, the surface over which the double integration extends. 

The value of the first term is seen (by what precedes) to be the 

vector-force due to uniform normal magnetisation of the same 



485.] Also, since VUp = — -^ > 

we obtain at once 

whence, by difierentiation, or by putting p + a for p, and expanding 
in ascending powers of Ta (both of which tacitly assume that the 
origin is external to the space integrated through, i.e., that Tp 
nowhere vanishes), we have 

and this, again, involves 

486.] The interpretation of these, and of more complex formulae 
of a similar kind, leads to many curious theorems in attraction and 
in potentials, Thus, from (1) of § 481, we have 


which.^ves the attraction of a mass of density t in terms of the 
potentials of volume distributions and surface distributions. Putting 

this becomes 



Tp JJJ Tp^ ~JJ Tp 
By putting cr = p, and taking the scalar, we recover a formula 
given above ; and by taking the vector we have 

r/fUvUpds = 0. 
This may be easily verified from the formula 

/Pdp = r//Uv.vPds, 

by remembering that VTp = Up. 

Again if, in the fundamental integral, we put 

(T = tUp, 

487.] As another application, let us consider briefly the Stress- 
function in an elastic solid. 

At any point of a strained body let A. be the vector stress per 
unit of area perpendicular to i, n and v the same for planes per- 
pendicular to J and k respectively. 

Then, by considering an indefinitely small tetrahedron, we have 
for the stress per unit of area perpendicular to a unit-vector <a the 
expression kSia) + iJ.SJ(o + vSko> =-<j>a>, 

so that the stress across any plane is represented by a linear and 
vector function of the unit normal to the plane. 

But if we consider the equilibrium, as regards rotation, of an 
infinitely small parallelepiped whose edges are parallel to i, j, k 
respectively, we have (supposing there are no molecular couples) 

F{iK+JlJ. + kv) = 0, 

or 2 Fi^i = 0, 

or r.V^p = 0. 

This shews (§173) that in this case (j) is self-conjugate, or, in other 
words, involves not nine distinct constants but only six. 

488.] Consider next the equilibrium, as regards translation, of 
any portion of the solid filling a simply-connected closed space. 
Let u be the potential of the external forces. Then the condition 
is obviously ff^ ( Vv) ds +fffdiVu = 0, 

where v is the normal vector of the element of surface ds. Here 

278 QUATEENIONS. [489- 

the double integral extends over the whole boundary of the closed 
space, and the triple integral throughout the whole interior. 

To reduce this to a form to which the method of § 467 is directly 
applicable, operate by S.a where a is any constant vector whatever, 
and we have /y S .(paUvds + yy/ds SaVu = 
by taking advantage of the self-conjugateness of (p. This may be 
written ///ds{S.V<t>a + 8.dVv.) = 0, 

and, as the limits of integration may be any whatever, 

8.V(t>a + S.aVu = (1) 

This is the required equation, the indeterminateness of a rendering 
it equivalent to tAree scalar conditions. 

There are various modes of expressing this without the a. Thus, 
if A be used for V when the constituents of <^ are considered, we 
may write Vu = -SVA.cjyp. 

In integrating this expression through a given space, we must 
remark that V and p are merely artificial symbols of construction, 
and therefore are not to be looked on as variables in the integral. 

489.] As a verification, it may be well to shew that from this 
equation we can get the condition of equilibrium, as regards rotation, 
of a simply connected portion of the body, which can be written 
by inspection as 

//r.p<p{Uv)ds+///r.pVuds = 0. 

This is easily done as follows : (1) gives 

S.V<t>(r + S.crVu = 0, 
if, and only if, <r satisfy the condition 
S4{V)(T = 0. 
Now this condition is satisfied if 

cr = Kap 
where a is any constant vector. For 

= S.aFV<t)p = 0. 
Hence ///'^s {S.Vcj) Fap + S.apVu) = 0, 

or f/dsS.apij}Uv+///ds S.apVu = 0. 

Multiplying by a, and adding the results obtained by making a in 
succession each of three rectangular vectors, we obtain the required 

490.] Suppose a- to be the displacement of a point originally at 
p, then the work done by the stress on any simply connected portion 
of the solid is obviously 



because <j) ( Uv) is the vector Toree overcome per unit of area on the 
element ds. This is easily transformed to 

491. J In this case obviously the strain-function is 
X (■nr) = ■ar — /S. (•srV)cr. 
Now if the strain be a mere rotation, in which case 

S.)(ZlT\T — S.-S7T = 0, 

whatever be the vectors ot and t, no work is done by the stress. 
Hence the expression for the work done by the stress must vanish 
if these conditions are fulfilled. 

Again, it is easily seen that when the strain is infinitely small 
the work must be a homogeneous function of the second degree of 
these critical quantities ; for, if it exist, it is essentially positive. 
Hence, even when finite, the work on unit-volume may be ex- 
pressed as » = 2.(5.x€X«'- -S""') {S-xrixn'-Sm'), 
where e, e', r), rf, which are in general functions of cr, become con- 
stant vectors if the stress is indefinitely small. When this is the 
case it is easy to see that, whatever be the number of terms under 
S, w involves twenty-one separate and independent constants only ; 
viz. the coefiicients of the homogeneous products of the second order 
of the six values of form 


iovthe values i, J, ^ of ot or r. 

Supposing the strain to be indefinitely small, we have for the 
variation of to, the expression 

.+ ^{S.x^X^'-S,e'){S.bxr,xri'+S.bxr,'xv)- 
Now, by the first equation, we have 

SxOT = — *S'(t!rV)8(r. 
Hence, writing the result for one of the factors only, the variation 
of the whole work done by straining a mass is 

bJr= b///wds =/ffbw & 

= -^fffd,{8.xy\xr\-Sm) {-S.xe'5.(€V)6<r-f-S.xe^(€'V)8<7}. 

Now, if we have at the limits 

8(7 = 0, 

i.e. if the surface of the mass is altered in a given way, we have 


fffdsS.'wS{€^)b<T = -///dsS.b(TS{iV)w. 

280 QUATERNIONS [492- 


Now any arbitrary change in o- will in general increase the amount 
of work done, so that we have 

= 2 [5(eV) {x«'('S.X'7X'?'-'Sw')} +'S(e'V) {xeC&XIX'?'-'^'/'?')}]. 
which is our equation for the determination of cr, as the constants 
e, i, t), rj' are dependent solely on the elastic properties of the sub- 
stance distorted, and may therefore be considered as known ; while 
X essentially involves o-. 

492.] Since the algebraic operator 

when applied to any function of a;, simply changes x into x-\-U, it 
is obvious that if o- be a vector not acted on by 
„ . d . d , d 
dx •' dy dz 
we have ,-s.vy(p) =/(p + ^), 

whatever function /"may be. From this it is easy to deduce Taylor's 
theorem in one important quaternion form. 

If A bear to the constituents of o- the same relation as V bears to 
those of p, and if_/and F be any two functions which satisfy the 
commutative law in multiplication, this theorem takes the curious 
form ,-^^^f{p) F{a) =/(p + A) F{<t) = F{<t + V)f{p) ; 
of which a particular case is 

,S^f(^,)F{y) =/{x + ±)FQ,) = F{y + ^)/W. 

The modifications which the general expression undergoes, when 
,/and i''are not commutative, are easily seen. 

If one of these be an inverse function, such as, for instance, may 
occur in the solution of a linear differential equation, these theorems 
of course do not give the arbitrary part of the integral, but they 
often materially aid in the determination of the rest. 

Other theorems, involving operators such as e*^, e^-'W^ &e., &c. 
are easily deduced, and all have numerous applications. 

493.] But there are among them results which appear startling 
from the excessively free use made of the separation of symbols. Of 
these one is quite sufficient to shew their general nature. 

Let P be any scalar function of p. It is required to find the 
difference between the value of P at p, and its mean value throughout 


a very small sphere, of radius r and volume v, whicli has the ex- 
tremity of p as centre. 

From what is said above, it is easy to see that we have the fol- 
lowing expression for the required result : — 

where o- is the vector joining the centre of the sphere with the ele- 
ment of volume <?s, and the integration (which relates to o- and & 
alone) extends through the whole volume of the sphere. Expanding 
the exponential, we may write this expression in the form 

higher terms being omitted on account of the smallness of r, the 
limit of T<T. 

Now, symmetry shews at once that 

fff^rd, = 0. 
Also, whatever constant vector be denoted by a, 
///{Sa^fds = -aV/f{S<rUafds. 
Since the integration extends throughout a sphere, it is obvious 
that the integral on the right is half of what we may call the 
moment of inertia of the volume about a diameter. Hence 

{8<TUafd^ = '"^^ 



If we now write V for a, as the integration does not refer to V, 
we have by the foregoing results (neglecting higher powers of r) 


which is the expression given by Clerk-Maxwell*. Although, for 
simplicity, P has here been supposed a scalar, it is obvious that in 
the result above it may at once be written as a quaternion. 

494.J If p be the vector of the element ds, where the surface 
density isfp, the potential at o- is 

F being the potential function, which may have any form whatever. 
By the preceding, § 492, this may be transformed into 

' London Math. Soc. Proc, vol. iii, no. 3^, 1871. 

282 QUATERNIONS. [495- 

or, far more conveniently for the integration, into 

where A depends on the constituents of a in the same manner as V 
depends on those of p. 

A still farther simplification may be introduced by using a vector 
a-Q, which is finally to be made zero, along with its corresponding 
operator Aq, for the above expression then becomes 

where p appears in a comparatively manageable form. It is obvious 
that, so far, our formulae might be made applicable to any distribu- 
tion. We now restrict them to a superficial one. 

495.] Integration of this last form can always be easily effected 
in the case of a surface of revolution, the origin being a point in 
the axis. For the expression, so far as the integration is concerned, 
can in that case be exhibited as a single integral 



where (f> may be any scalar function, and x depends on the cosine of 
the inclination of p to the axis. And 

As the interpretation of the general results is a little troublesome, 
let us take the case of a spherical shell, the origin being the centre 
and the density unity, which, while simple, sufficiently illustrates 
the proposed mode of treating the subject. 

We easily see that in the above simple case, a being any constant 
vector whatever, and a being the radius of the sphere, 

/"+" 2 Tra 

J —a ■^<* 

Now, it appears that we are at liberty to treat A as a has just been 
treated. It is necessary, therefore, to find the effects of such opera- 
tors as TA, e"^'^, &c., which seem to be novel, upon a scalar function 
of To- ; or %, as we may for the present call it. 


Now (rA)2i?'=-A2J = 2?"' + — , 

whence it is easy to guess at a particular form of TA. To be sure 
that it is the only one, assume 


where </> and i|f are scalar functions of JC to be found. This gives 

= 4>^F" + (<^^' 4- v/'<#) + <i>i') F' + {<pf' + ^^) F. 
Comparing, we have 


(^\/f' + i|'^ = 0. 
From the first, ^ = ± Ij 

whence the second gives '>/' = + — > 

the signs of ^ and \/f being alike. The third is satisfied identically. 

That is +yA = ^ + -- 

~ a® St 

Also, an easy induction shews that 

±(»)- = (a)"+5(»r 

Hence we have at once 

by the help of which we easily arrive at the well-known results. 
This we leave to the student*. 

496.] As an elementary example of the use of V in connection 
with the Calculus of Variations, let us consider the expression 

A =/QTdp, 
where Tdp is an element of a finite are along which the integration 
extends, and Q is in general a scalar function of p and constants. 
We have bA z=/{bQTdp+QbTdp) 

=/{bQTdp- QS. Udpdbp) 
= -iqSUdphp-] +/{bQTdp+S.bpdiQUdp)), 
where the portion in square brackets refers to the limits only, and 
gives the terminal conditions. The remaining portion may easily 
be put in the form 


* Proc. B. S. E., 1871-2. 

284 QUATERNIONS. [497- 

If the curve is to be determined by the condition that the varia- 
tion of A shall vanish, we must have, as 8p may have any direction, 

or, with the notation of Chap. IX, 

This simple equation shews that 

(1) The osculating plane of the sought curve contains the 
vector VQ. 

(2) The curvature at any point is inversely as Q, and directly as 
the component of V Q parallel to the radius of absolute curvature. 

497.] As a first application, suppose A to represent the action of 
a particle moving freely under a systan of forces which have a 
potential*, so that Q := ^o, 

and p2 = 2 {P-H), 

where P is the potential, H the energy constant. 
These give TpVTp = QVQ = -VP, 

and qp'= p, 

so that the equation above becomes simply 

p + VP = 0, 
which is obviously true. 

498.] If we look to the superior limit only, the first expression 
for 6^ becomes in the present case 

-{TpSUdptp'] = -Sphp. 

If we suppose a variation of the constant H, we get the following 
term from the unintegrated part 


Hence we have at once Hamilton's equations of varying action in 
the forms y^ _ a 

and ^ = t. 

The first of these gives, by the help of the condition above, 
(VJ)2 = 2 {P-H), 

the well-known partial difierential equation of the first order and 
second degree. 

499.] To shew that, if A be any solution whatever of this equa- 
tion, the vector VA represents the velocity in a free path capable of 


being described under the acflon of the given system of forces, we 

-j^P = P =-VP=-\V{VAf 
= ~S{VA.V)VA. 
But ~'VA=-S{fiV)VA. 

A comparison shews at once that the equality 

VA = p 
is consistent with each of these vector equations. 
500.] Again, if 5 refer to the constants only, 
J a(VJ)2 = S.VA1>VA =-lH 
by the differential equation. 

But we have also — - = t, 

which gives 17^-^) — — 'S'(pV)aJ = 'dH. 

These two expressions for 3 jy again agree in giving 

VA = p, 
and thus shew that the differential coefficients of A with regard to 
the two constants of integration must, themselves, be constants. 
We thus have the equations of two surfaces whose intersection 
determines the path. 

501.] Let us suppose next that A represents the time of passagCj 
so that the brachistochrone is required. Here we have 

the other condition being as in § 497, and we have 

which may be reduced to the symmetrical form 

p+p-^VP/J = 0. 
It is very instructive to compare this equation with that of the free 
path as above, § 497. 

The application of Hamilton's method may be easily made, as in 
the preceding example. (Tait^ Trans. R. S. E., 1865.) 

503.] As a particular case, let us suppose gravity to be the only 
force, then VP = a, 

a constant vector, so that 

286 QUATERNIONS. [503- 

The form of this equation suggests the assumption 

where jo and q are scalars and 

Sap = 0. 
Substituting, we get 

-j)qseo^qt + {-P'>-p'^a^ian^qt) = 0, 
which gives joq = T^^ = p^T^a. 

Now let jo /3~^o = y ; 

this must be a unit-vector perpendicular to a and /3, so that 

ir^ = -^— , (cos at— yBm at), 
cosqt '• ^ • 

whence p = cos qt {cos qt + y sin qt)P~^ 

(which may be verified at once by multiplication). 

Finally, taking the origin so that the constant of integration 

may vanish, we have 

2/3/8 = t+ — (siQ2g'^— ycos22'^), 

which is obviously the equation of a cycloid referred to its vertex. 
The tangent at the vertex is parallel to /3j and the axis of symmetry 
to a. 

503.] In the case of a chain hanging under the action of given 
forces Q = Pr, 

where P is the potential, r the mass of unit-length. 

Here we have also, of course, 

/Tdp = I, 
the length of the chain being given. 

It is easy to see that this leads, by the usual methods, to the 

equation -=- {{Pr + ii)p'} —rVP = 0, 

where u is a scalar multiplier. 

504.J As a simple case, suppose the chain to be uniform. Then 
r may be merged in u. Suppose farther that gravity is the only 
force, then P = Sap, VP = —a, 

and -J- {{Sap+u)p'} +a = 0. 

Differentiating, and operating by Sp\ we find 

S.p'[p'{8ap'+^)+a'^ = 0; 

which shews that u is constant, and may therefore be allowed for 
by change of origin. 


The curve lies obviously in% plane parallel to a, and its equation 
is {8apY + a^ s^ = const., 

which is a well-known form of the equation of the catenary. 

When the quantity Q of § 496 is a vector or a quaternion, we 
have simply an equation like that there given for each of the con- 

505.] Suppose P and the constituents of a- to be functions which 
vanish at the bounding surface of a simply-connected space 2, or 
such at least that either P or the constituents vanish there, the 
others (or other) not becoming infinite. 

Then, by § 467, 

///d,S.V{Pa) =//dsPSaUv = 0, 
if the integrals be taken through and over 2. 

Thus ///dsS.(rVP = -///dsPS.V<T. 

By the help of this expression- we may easily prove a very re- 
markable proposition of Thomson {Cam. and Dub. Maih. Journal^ 
Jan. 1848, or Reprint of Papers on Electrostatics, § 206.) 

To shew that there is one, and lut one, solution of the equation 
S.V{e^Vu)= 4ir>- 
where r vanishes at anminfinite distance, and e is any real scalar what- 
ever, continuous or discontinuous. 

Let V be the potential of a distribution of density r, so that 

V^v = 4 nr, 
and consider the integral 

q = —JjJ^s (eVu- -Vv) . 

That Q may be a minimum as depending on the value of u (which 
is obviously possible since it cannot be negative, and since it may 
have any positive value, however large, if only greater than this 
minimum) , we must have 

= ibQ =-///dsS.(e^Vu—Vv)Vbu 
= ///<^s bu S.V {e^Vu-Vv), 
by the lemma given above, 

=/y/dsbti {S.V {e^Vu)-4:T!r}. 
Thus any value of u which satisfies the given equation is such as to 
make Q a minimum. 

But there is only one value of w which makes Q a minimum ; 
for, let Qi be the value of Q when 

«j^ = w + (^ 
is substituted for this value of u, and we have 

288 QUATERNIONS. [505- 

Qi = —JJJds. (eV (m + <^) - i V w) 

The middle term of this expression may, by the proposition at the 
beginning of this section, be written 

and therefore vanishes. The last term is essentially positive. Thus 
if % anywhere differ from u (except, of course, by a constant quan- 
tity) it cannot make Q a, minimum ; and therefore m is a unique 


1. The expression 

Fo/3 Fyb + Fay Vh^ + TaS V^y 
denotes a vector. What vector ? 

2. If two surfaces intersect along a common line of curvature, 
they meet at a constant angle. 

3. By the help of the quaternion formulae of rotation, translate 
into a new form the solution (given in § 234) of the problem of 
inscribing in a sphere a closed polygon the directions of whose sides 
are given. 

4. Express, in terms of the masses, and geocentric vectors of the 
sun and moon, the sun's vector disturbing force on the moon, and 
expand it to terms of the second order; pointing out the mag- 
nitudes and directions of the separate components. 

(Hamilton, Lectures, p. 615.) 

5. J£ q = r^, shew that 

2dq = 2dri = i {dr+Kqdrq-^)Sq-^ = i {dr + q-^drKq)Sq-'- 

= (drq + Kqdr)q-''{q + Kq)-^ = {drq + Kqdr){r+Tr)-^ 

_ dr+Uq-^drUq-^ _ drUq + Uq-^dr _ q-^{U'qdr + drUq-'^) 
~ Tq{Uq+Uq-^) ~ q{Uq+Uq-^) " Uq+Uq-^ 

_ q-^{qdr + Trdrq~'^) _ drUq-{- Uq-^d^ _ drKq-^ +q-'^dr 
~ Tq{Uq+Uq-^) ~ Tq{l + Ur) " iTUr 



2clq =^\clr+ r.Fdrjqlq-^ = j 3r -V.Vdrj q-'' \q- 

q q S^ q q S ^ 

= drq-^ + V. Vq-^ Vcl/r (l + -^ j-i) : 

and give geometrical interpretations of these varied expressions for 
the same quantity. {Ihid. p. 628.) 

6. Shew that the equation of motion of a homogeneous solid of 
revolution about a point in its axis, which is not its centre of 
gravity, is BYp^-ASlp = Ypy, 

where 12 is a constant. {Trans. U. 8. E., 1869.) 

7. Integrate the differfential equations : 

{a.) % + aq = h, 

where a and h are given quaternions, and and -v/f given linear and 
vector functions. (Tait, Proe. B.S.E., 1870-1.) 

8. Derive (4) of § 92 directly from (3) of § 91. 

9. Find the successive values of the continued fraction 

where i and j have their quaternion significations, and so has the 
values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.) 

10. If we have m. = f-A) c, 

where c is a given quaternion, find the successive values. 

For what values of c does u become constant ? {Ihid. p. 652.) 

11. Prove that the moment of hydrostatic pressures on the faces 
of any polyhedron is zero, {a.) when the fluid pressure is the same 
throughout, {b.) when it is due to any set of forces which have a 

12. What vector is given, in terms of two known vectors, by the 
relation p-^ = \ {ar^ + yS'^) ? 

Shew that the origin lies on the circle which passes through the 
extremities of these three vectors. 



13. Tait, Tram, and Proc. R.S.B., 1870-3. 
With the notation of §§ 467, 477, prove 

(«•) ///S{aV)rds =//rSaUvds. 
(6.) I{ S{pV)T = -nT, 

(« + 3)///r& = -f/rSp Uvds. 

(e.) With the additional restriction V^r = 0, 

//S.mi2np+{n+3)p^V).Tds = 0. 

(d.) Express the value of the last integral over a non- 
closed surface by a line-integral. 

(e.) -/Tdp =f/ds8.UvV<T, 

if (7 = Udp all round the curve. 

{/.) For any portion of surface whose bounding edge lies 
wholly on a sphere with the origin as centre 

ffds8.{UpUvV).<r = 0, 
whatever be the vector o-. 

iff.) /rdpV.tr =//ds{UvV^-S{UvV)V)(T, 
whatever be o-. 

14. Tait, Trans. B. S. U., 1873. 
Interpret the equation 

d(T = uqdpq~^, 
and shew that it leads to the following results 

V^cr = qVn q~^, 
V.Mj-i = 0, 
V^M* = 0. 
Hence shew that the only sets of surfaces which, together, cut 
space into cubes are planes and their electric images. 

1 5. What problem has its conditions stated in the following six 
equations, from which ^, rj, ( are to be determined as scalar functions 
ot x, y, g, or oi p = is!+jy+kz'> 

V^i = 0, V^r, = 0, V^f = 0, 

SViVrj = 0, SVriVC= 0, SV^Vi = 0, 

„ . d . d , d 
where V = »^- + ?^- +/e-=- ■ 
dx '' dy dz 

Shew that they give the farther equations 


Shew that (with a change OT origin) the general solution of these 
equations may be put in the form 

where <j(> is a self-conjugate linear and vector function, and £, rj, ( 
are to be found respectively from the three values of_/at any point 
by relations similar to those in Ex. 24 to Chapter IX. (See Lame, 
Journal de MatAematiqties, 1843.) 

16. Shew that, if p be a planet's radius vector, the potential P of 
masses external to the solar system introduces into the equation of 
motion a term of the form S (pV)VP. 

Shew that this is a self-conjugate linear and vector function 
of p, and that it involves only Jive independent constants. 

Supposing the undisturbed motion to be circular, find the chief 
effects which this disturbance can produce. 

17. In § 405 above, we have the equations 

?a(OT + «^OT) = 0, 8a^ =0, d = aiFia, Ta = 1, 
where u>^ is neglected. Shew that with the assumptions 

bit Uif 

qz^i", a = qPq-'^, r = fi", •sr = qrrr-^q-^, 
we have /3 = 0, Tj3 = 1, S/3t=0, F0{T + n^T) = O, 

provided co*S«a— coj^ = 0. Hence deduce the behaviour of the Fou- 
cault pendulum without the x, y, and ^, jj transformations in the 

Apply analogous methods to the problems proposed at the end of 
§ 401 of the text. 

18. Hamilton, Bishop Law's Premmm Examination, 1862. 

[a.) If OABP be four points of space, whereof the three first are 
given, and not eoUinear ; if also oa = a, ob = /3, op = p ; 
and if, in the equation 

a a 

the characteristic of operation F be replaced by S, the 

locus of P is a plane. What plane ? 
{i.) In the same general equation, if F be replaced by V, the 

locus is an indefinite right line. What line ? 
(c.) If F be changed to K, the locus of p is a point. What 

point ? 
(d.) If F be made = TJ, the locus is an indefinite half-line, or 

ray. What raj^ ? 


(e.) If F be replaced by T, the locus is a sphere. What sphere ? 

{/.) If F be changed to TV, the locus is a cylinder of revo- 
lution. What cylinder ? 

{g.) If 2?' be made TVU, the locus is a cone of revolution. What 
cone ? 

[h.) If SU be substituted for F, the locus is one sheet of such a 
cone. Of what cone ? and which sheet ? 

(«.) If i'' be changed to VU, the locus is a pair of rays. Which 

19. Hamilton, Bishop Law's Premium Examination, 1863. 

{a.) The equation Spp' + a^ — 

expresses that p and p' are the vectors of two points 
p and p', which are conjugate with respect to the sphere 

or of which one is on the polar plane of the other. 

(b.) Prove by quaternions that if the right line pp', connecting 
two such pointSj intersect the sphere, it is cut har- 
monically thereby. 

(c.) If p' be a given external point, the cone of tangents drawn 
from it is represented by the equation, 

irppy = a^p-py; 

and the orthogonal cone, concentric with the sphere, by 

i8ppy+a^p"' = 0. 

{d.) Prove and interpret the equation, 

T{np-a) = T{p-na\ if Tp = Ta. 

{e.) Transform and interpret the equation of the ellipsoid, 

y(tp + p/() = K2_t2. 

{/.) The equation 

{k^-I^Y = {l^ + K^)Spp' + 2SLpKp' 

expresses that p and p' are values of conjugate points, 
with respect to the same ellipsoid. 

(ff.) The equation of the ellipsoid may also be thus written, 

S,;p = 1, if {k'^-l^)^v = {i.-kYp+2iSkp+ 2kSip. 
{h.) The last equation gives also. 


{i.) With the same sigiltfication of v, the differential equations 
of the ellipsoid and its reciprocal become 
Svdp — 0, Spdu = 0. 

{j.) Eliminate p between the four scalar equations, 
Sap = a, Spp = b, Syp = c, Sep = e. 

20. Hamilton, Bishop Law^s Premium Examination, 1864. 

{a.) Let Aj^B-^j A^,^^, ... A^B„ be any given system of posited 
right lines, the 2n points being all given; and let 
their vector sum, 

AB = Aj^B^+A^B^+.-.+A^B^, 
be a line which does not vanish. Then a point H, and 
a scalar A, can be determined, which shall satisfy the 
quaternion equation, 

HAj^.A^Bi+... +HA^.A^B^ = h.AB ; 
namely by assuming any origin 0, and writing, 

Qjj_ jr OA-AA + ■ ■ • + OAn-A„B„ 

A^B,+ ... 

(b.) For any assumed point C, let 

Qc = CA^.A^B^ + . . . + CA^.A^B,, ; , 
then this quaternion sum may be transformed as follows, 

Qc= Qh + CH.AB = {7i + GH).AB ; 
and therefore its tensor is 

Tqc = {fi'' + CH^f.lB, 
in which AB and CH denote lengths. 

(c.) The least value of this tensor TQc is obtained by placing 
the point. C at H; if then a quaternion be said to be a 
minimum when its tensor is such, we may write 

min. Qc = Qj=r= h.AB; 
so that this minimum of Qc is a vector. 

{d.) The equation 

TQc = c = any scalar constant > TQh 
expresses that the locus of the variable point C is a 
spheric surface, with its centre at the fixed point H, 
and with a radius r, or CH, such that ' 

r.AB = {TQc^-TQH^)i = (c^ - h\ AB^)^ ; 


so that H, as being thus the common centre of a series 
of concentric spheres, determined by the given system 
of right lines, may be said to be the Central Point, or 
simply the Centre, of that system. 

(e.) The equation 

TFQc = Cj = any scalar constant > TQh 
represents a right cylinder, of which the radius 

divided by AB, and of which the axis of revolution is 
the line, VQc = Qh = h.AB; 

wherefore this last right line, as being the common 
axis of a series of such right cylinders, may be called 
the Central Axis of the system. 

(/".) The equation 

SQc = ^2 = ^°y scalar constant 
represents a plane ; and all such planes are parallel to 
the Central Plane, of which the equation is 

{g.) Prove that the central axis intersects the central plane 

perpendicularly, in the central point of the system. 


(Ji,.) When the n given vectors A^B-i, ... A„B„ are parallel, and 
are therefore proportional to n sealars, b^,...6„, the 
scalar A and the vector Qh vanish ; and the centre H is 
then determined by the equation 

bi.HAi+i2SA+--- + h-SA„= 0, 
or by the expression, 

where is again an arbitrary origin. 
21. Hamilton, Bishop Law's Premium Examination, 1860. 

{a.) The normal at the end of the variable vector p, to the 
surface of revolution of the sixth dimension, which is 
represented by the equation 

(p2-a2)3 = 27a==(p-a)*, (a) 

or by the system of the two equations, 

p2_a2=3<2„2^ (p_„)2 = ^3^2^ ^^,^ 


and the tangent to the meridian at that point, are 
respectively parallel to the two vectors, 

• v = 2{p-a)-tp, 
and T=2{l-2i){p-a) + i^p; 

so that they intersect the axis a, in points of which the 
vectors are, respectively, 

2a 2{l — 2f)a 

2-t' -^"^ {2-tf-2 


If dp be in the same meridian plane as p, then 

t{l t){i-f)dp=3Tdi, and s''f* =^~* 

up 3 

Under the same condition. 


'^^ = 1(1-^). 
dp 3^ ' 

(d.) The vector of the centre of curvature of the meridian, at 
the end of the vector p, is, therefore, 

/„<?Dx-^ 3 V 6a — (4 — Op 

'' = P-<^dp) =P-2Y^t= 2(1-0 • 

{e.) The eiqjressions in Example 38 give 

v^ = a?t^{\-tf, T^ = aH%l-f)^4:-f); 

9 9 a^t 
hence (cr—pY = -t^P', and dp''' = -.dt"^; 

the radius of curvature of the meridian is, therefore. 

and the length of an element of arc of that curve is 

= Tdp=zTa{-^fdt. 

{/.) The same expressions give 

thus the auxiliary scalar t is confined between the limits 
and 4, and we may write t = 2 vers $, where 5 is a 
real angle, which varies continuously from to 2Tr ; the 
recent expression for the element of arc becomes, there- 
fore, ds=3TaJd0* 
and gives by integration 

s = 6Ta{e-sme), 
if the arc s be measured from the point, say F, for which 
p = a, and which is common to all the meridians ; and 
the total periphery of any one such curve is = 12Tr Ta. 


{ff.) The value of o- gives 

i{<r^-a^) = 3aH{i-t), 16{Fa<r)^ = -a*f{i-tf ; 
if, then, we set aside the axis of revolution o, which is 
crossed by all the normals to the surface (a), the surface 
of centres of curvature which is touched by all those 
normals is represented by the equation, 

4 (0-2 -a2)3 + 27 a2(rao-)2 = (b) 

{h.) The point F is common to the two surfaces (a) and (b), 
and is a singular point on each of them, being a triple 
point on (a), and a double point on (b) ; there is also at 
it an infinitely sharp cusp on (b), which tends to coincide 
with the axis a, but a determined tangent plane to (a), 
which is perpendicular to that axis, and to that cusp ; 
and the point, say.?", of which the vector =— a, is 
another and an exactly similar cusp on (b), but does not 
belong to (a). 

(j.) Besides the three universally coincident intersections of the 
surface (a), with any transversal, drawn through its 
triple point F, in any given direction y9, there are 
always three other real intersections, of which indeed one 
coincides with F if the transversal be perpendicular to 
the axis, and for which the following is a general 
formula : 

p=Ta.[Ua+ {28U{a^)iYU^']. 

{j.) The point, say V, of which the vector is p=2a, is a 
double point of (a), near which that surface has a cusp, 
which coincides nearly with its tangent cone at that 

point ; and the semi-angle of this cone is = - • 

Auxiliary Equations : 

(2Sp{p—a) = aH^{3 + t), 
\2Sa{p-a) = aH^ls-t). 
f Svp =-aH{l—t){l—2t), 
l2Sv{p-a) = aH''{l-f). 
( SpT^aH^{l~t){^-t), 
l2S(p-a)T= aH^l-t){4:-t). 

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2 vols. Demy Ocftavo. i^s. 

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With a Metrical Translation, Notes and Introduction, by E. H. 
Palmer, M.A., Barrister-at-Law of the Middle Temple, Lord 
Almoner's Professor of Arabic and Fellow of St John's College 
in the University of Cambridge, 3 vols. Crown Quarto. 

Vol. I. The Arabic Text. ioj. 6d. \ Cloth extra, 15J. 

Vol. II. English Translation, ioj. 6^/.; Cloth extra, 15 j. 

"Professor Palmer's activity in advancing 
Arabic scholarship has formerly shown itself 
in the production of his excellent Arabic 
Grammar, and his Descriptive Catalogue of 
Arabic MSS. in the Library of Trinity Col- 
lege, Cambridge. He has row produced an 
admirable text, which illustrates in a remark- 
able manner the flexibility and graces of the 
language he loves so well, and of which he 
seems to be perfect master.... The Syndicate 
of Cambridge University must not pass with- 
out the recognition of their liberality in 
bringing out, in a worthy form, so important 
an Arabic text. It is not the first time that 
Oriental scholarship has thus been wisely 
subsidised by Cambridge." — Indian Mail. 

"It is impossible to quote this edition with- 
out an expression of admiration for the per- 
fection to which Arabic typography has been 
brought in England in this magnificent Ori- 
ental work, the production of which redo,unds 
to the imperishable credit of the University 
of Cambridge. It may be pronounced one of 
the most beautiful Oriental books that have 
ever been printed in Europe : and the learning 
of the Editor worthily rivals the technical 
get-up of the creations of the soul of one of 
the most tasteful poets of Islftm, the study 
of which will contribute not a little to save the 
honour of the poetry of the Arabs. Here 
first we make the acquaintance of a poet who 
grives us something better than monotonous 
descriptions of camels and deserts, and may 
even be regarded as superior in charm to al 
Mutanabbi." — Mythologv among the He- 
brews {Engl. TransL), p. 194. ' 

"Professor Palmer has produced the com- 
plete works of Behi-ed-din Zoheir in Arabic, 
and has added a second volume, containing 
an .English verse translation of the whole. 

It is only fair to add that the book, 

by the taste of its arabesque binding, as well 
as by the beauty of the typography, which 
reflects great credit on the Cambridge Uni- 
versity Press, is entitled to a place in the 
drawing-room." — Times. 

"For ease and facility, for variety of 
metre, for imitation, either designed or un- 
conscious, of the style of several of our own 

poets, these versions deserve high praise 

We have no hesitation in saying that in both 
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grateful ; and that, while his knowledge of 
Arabic is a sufficient guarantee for his mas- 
tery of the original, his English compositions 
are distinguished by versatility, command of 
language, rhythmical cadence, and, as we 
have remarked, by not unskilful imitations of 
the styles of several of our own favourite 
poets Hying and dtvid.."— Saturday Review. 
"This sumptuous edition of the poems of 
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tion to the small series of Eastern poets' 
accessible to readers who are not Oriental- 
ists. ... In all there is that exquisite finish of 
which Arabic poetry is susceptible in so rare 
a degree. The form is almost always beau- 
tiful, be the thought what it may. But this, 
of course, can only be fully appreciated by 
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lation. It. is excellently well done. Mr 
Palmer has tried to imitate the fall of the 
original in his selection of the English metre 
for the various pieces, and thus contrives to 
convey a faint idea of the graceful flow of 

the Arabic Altogether the inside of the 

book is worthy of the beautiful arabesque 
binding that rejoices the eye of the .lover of 
Arab axt."-— Academy. 

containing the Sanskrit Text in Roman Characters, followed by a 
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to derived words in Cognate Languages, and a sketch of Sanskrit 
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Regius Professor of Hebrew, late Professor of Arabic, and formerly 
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GREEK AND LATIN CLASSICS, &c. (See also pp. 20-23.) 


With a Translation in English Rhythm, and Notes Critical and Ex- 
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" One of the best editions of the master- tion of a great undertaking." — Sat. Rem, _ 
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" By numberless other like happy and ble piece of the highest criticfsm. _. . . . Hikft 

weighty helps to a coherent and consistent your Preface extremely ; it is' just to the 

text and interpretation, Dr Kennedy has point." — Professor Paley. 
approved himself a guide to Aeschylus of " Professor Kennedy has conferred a boon 

certainly peerless calibre." — Contemp. Rev. on all teaphers of the Greek classics, by caus- 

"Itis needless to multiply proofs of the ing the substance of his lectures at Cam- 
value of this volume alike to the poetical bridge on the Agamemnon of ^schylus to 
translator, the critical scholar, and the ethical be pubIished...This edition of the Agamemnon' 
student. We must be contented to thank is one which no classical master should be 
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ARISTOTLE. Edited by Henry Jackson, M.A., Fellow of Trinity 
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with Introductions and English Notes, by F. A. Paley, M.A. Editor 
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accurate, his experience of editing wide, and the elucidation of matters of daily life, inthe 

if he is content to devote his learning and delineation of which Demosthenes is so rich, 

abilities to the production of such manuals obtains full justice at his hands We 

as these they will be received with gratitude hope this edition may lead the way to a more 

throughout the higher schools of the country. general study of these speeches in schools 

Mr ^dys is deeply read in the German than has hitherto been ^ossihle.— Academy. 

Part II.' Pro Phormione, Contra Stephanum I. II. ; Nicostratum, 
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■Umits though we can hardly conceive a task upon a work cvmcrag more pams, scho ar- 

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T f Is a great boon to those who set them- Orations of Demosthenes . —Sat. Rev 
selves to unravel the thread of arguments ". ..... the edition reflects credit on 

nro and con to have the aid of Mr Sandys's Cambridge scholarship, and ought to be ex- 
excellent running commentary . . . .and no tenslvely used. '-Athen,Bum. 

one can say that he is- ever deficient 

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"Considered simply as a contribution to 
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it has a wider interest, as exemplifying the 
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literally translated, by the late E. M, COPE, Fellow of Trinity College, 
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THE RHETORIC. With a Commentary by the late E. M. CoPE, 
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Syndics of the University Press by J. E. Sandys, M.A., Fellow and 
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a biographical jyiemoir by H. A. J. MUNRO, M.A. Three Volumes, 
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not equally accompanied by those qualities of 
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should not have received the last touches of 
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Mr Sandys, for the manly, unselfish, and un- 
flinching spirit in which he has performed his 
most difficult and delicate task. If an English 
student wishes to have a full conception of 
what is contained in the RJietoric of Aris- 
totle, to Mr Cope's edition he must go."— 

" Mr Sandys has performed his arduous 
duties with marked ability and admirable tact. 
...Besides the revision of Mr Cope's material 
already referred to in his own words, Mr 
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carefully supplies 'the deficiency, following 
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best of the late Mr Shilleto's 'Adversaria.* 
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* ' A careful examination of the work shows 
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scholarship, and above all, that sound judg- 
ment and never-failing good sense which are 
the crowning merit of our best English edi- 
tions of the Classics, all combine to make 
this one of the most valuable additions to the 
knowledge of Greek literature which we have 
had for many years," — Spectator. 

^ "Von der Rhetorik isteineneue Ausgabe 
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with Marginal Analysis, an English Commentary, and copious Indices, 
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Libri Tres, with Introduction and Commentary by Joseph, B. Mayor, 
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with a new collation of several of the English MSS. by J. H. Swain- 
SON, M.A., formerly Fellow of Trinity College, Cambridge. 

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Written between 1771 and 1781, Edited from the original manuscripts 
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some of them devoted to mathematical dis- Maxwell to see this goodly volume completed 

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not only the first and the greatest book on to physical phenomena, and is remarkable for 
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ever ejcpert." — Extract from letter of Pro- key which first opened to them the treasure- 

fessor Clerk Maxwell. house of mathematical physics. It is still the 

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troduccd to those English students of Mathe- half a century old." — Nature. 

By George Gabriel Stokes, M.A., D.C.L., LL.D., F.R.S., Fellow 
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the Roman Law at Cambridge which is now student will be interested as well as perhaps 

so marked a feature in the industrial life surprised to find how abundantly the extant 

of the University. ■. . . In the present book fragments illustrate and clear up points which 

we have the fruits of the same kind of have.attracted his attention in the Commen- 

thorough and well-ordered study which was taries, or the Institutes, or the Digest."— 

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- OF ULPIAN. (New Edition, revised and enl^-rged.) 
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"As scholars and as editors Messrs Abdy "The number of books on various subjects 

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Gaius to the reader with few notes and those ^ genuine and increasing. The present edition 

merely by way of reference or necessary of Gaius and Ulpian from the Cambridge 

explanation. Thus the Roman jurist is University Press indicates that the Universi- 

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fecls that he is really studying Roman law ment."— Zaw youmaL 
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translated with Notes by J. T. Abdy, LL.D., Judge of County Courts, 
late Regius Professor of Laws in the University of Cambridge, and 
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is wanted. This translation will then be of a book which, is both elegant and useful."— 

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Part II. Da Adquirendo rerum dominio and De Adquirenda velamit- 
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' Demy 8vo. 48J. 

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a time when, our historians would raise the 
cry of protection for native industry. Of 
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Great,_ Goethe and Stein — the first two found 
long since in Carlyle and Lewes biographers 
who have undoubtedly driven their German 
competitors out of the field. And now in the 
year just past Professor Seeley of Cambridge 
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whichj though it modestly declines competi- 
tion with German w6rks and disowns the 
presumption of teaching us Germans our own 
history, yet casts into the shade by its bril- 
liant superiority all that we have ourselves 
hitherto written about Stein..,. In five long 
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as the late Mr Lewes and Professor Seeley, 

doing for German as well as English readers 
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us." — Times. 

" In a notice of this kind scant justice can 
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England that on the especial field of the Ger- 
mans, history, on the history of their own 
country, by the use of their own literary 
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the University during the troublous times of 
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"We trust Mr Mullinger will yet continue 
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" The, work displays very wide reading, 
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translated by MuNSHi Shew Shunker Singh and Pandit Shri 
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